Metamath Proof Explorer


Theorem mptbi12f

Description: Equality deduction for maps-to notations. (Contributed by Giovanni Mascellani, 10-Apr-2018)

Ref Expression
Hypotheses mptbi12f.1 𝑥 𝐴
mptbi12f.2 𝑥 𝐵
Assertion mptbi12f ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥𝐴 𝐷 = 𝐸 ) → ( 𝑥𝐴𝐷 ) = ( 𝑥𝐵𝐸 ) )

Proof

Step Hyp Ref Expression
1 mptbi12f.1 𝑥 𝐴
2 mptbi12f.2 𝑥 𝐵
3 1 2 nfeq 𝑥 𝐴 = 𝐵
4 eleq2 ( 𝐴 = 𝐵 → ( 𝑥𝐴𝑥𝐵 ) )
5 3 4 alrimi ( 𝐴 = 𝐵 → ∀ 𝑥 ( 𝑥𝐴𝑥𝐵 ) )
6 ax-5 ( ( 𝑥𝐴𝑥𝐵 ) → ∀ 𝑦 ( 𝑥𝐴𝑥𝐵 ) )
7 5 6 sylg ( 𝐴 = 𝐵 → ∀ 𝑥𝑦 ( 𝑥𝐴𝑥𝐵 ) )
8 eqeq2 ( 𝐷 = 𝐸 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) )
9 8 alrimiv ( 𝐷 = 𝐸 → ∀ 𝑦 ( 𝑦 = 𝐷𝑦 = 𝐸 ) )
10 9 ralimi ( ∀ 𝑥𝐴 𝐷 = 𝐸 → ∀ 𝑥𝐴𝑦 ( 𝑦 = 𝐷𝑦 = 𝐸 ) )
11 df-ral ( ∀ 𝑥𝐴𝑦 ( 𝑦 = 𝐷𝑦 = 𝐸 ) ↔ ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑦 ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) )
12 10 11 sylib ( ∀ 𝑥𝐴 𝐷 = 𝐸 → ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑦 ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) )
13 19.21v ( ∀ 𝑦 ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ↔ ( 𝑥𝐴 → ∀ 𝑦 ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) )
14 13 albii ( ∀ 𝑥𝑦 ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ↔ ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑦 ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) )
15 12 14 sylibr ( ∀ 𝑥𝐴 𝐷 = 𝐸 → ∀ 𝑥𝑦 ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) )
16 id ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) )
17 16 alanimi ( ( ∀ 𝑦 ( 𝑥𝐴𝑥𝐵 ) ∧ ∀ 𝑦 ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ∀ 𝑦 ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) )
18 17 alanimi ( ( ∀ 𝑥𝑦 ( 𝑥𝐴𝑥𝐵 ) ∧ ∀ 𝑥𝑦 ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ∀ 𝑥𝑦 ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) )
19 7 15 18 syl2an ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥𝐴 𝐷 = 𝐸 ) → ∀ 𝑥𝑦 ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) )
20 tsan2 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( 𝑥𝐴 ∨ ¬ ( 𝑥𝐴𝑦 = 𝐷 ) ) )
21 20 ord ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ 𝑥𝐴 → ¬ ( 𝑥𝐴𝑦 = 𝐷 ) ) )
22 tsbi2 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ( ( 𝑥𝐴𝑦 = 𝐷 ) ∨ ( 𝑥𝐵𝑦 = 𝐸 ) ) ∨ ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) )
23 22 ord ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ( ( 𝑥𝐴𝑦 = 𝐷 ) ∨ ( 𝑥𝐵𝑦 = 𝐸 ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) )
24 23 a1dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ( ( 𝑥𝐴𝑦 = 𝐷 ) ∨ ( 𝑥𝐵𝑦 = 𝐸 ) ) → ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) ) )
25 ax-1 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ( ( 𝑥𝐴𝑦 = 𝐷 ) ∨ ( 𝑥𝐵𝑦 = 𝐸 ) ) → ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) ) )
26 24 25 contrd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ∨ ( 𝑥𝐵𝑦 = 𝐸 ) ) )
27 26 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ 𝑥𝐴 → ( ( 𝑥𝐴𝑦 = 𝐷 ) ∨ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) )
28 21 27 cnf1dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ 𝑥𝐴 → ( 𝑥𝐵𝑦 = 𝐸 ) ) )
29 simplim ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) )
30 29 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ( 𝑥𝐴 ∨ ¬ 𝑥𝐵 ) → ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) ) )
31 tsbi3 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴 ∨ ¬ 𝑥𝐵 ) ∨ ¬ ( 𝑥𝐴𝑥𝐵 ) ) )
32 31 ord ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ( 𝑥𝐴 ∨ ¬ 𝑥𝐵 ) → ¬ ( 𝑥𝐴𝑥𝐵 ) ) )
33 tsan2 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑥𝐵 ) ∨ ¬ ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) ) )
34 33 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ( 𝑥𝐴 ∨ ¬ 𝑥𝐵 ) → ( ( 𝑥𝐴𝑥𝐵 ) ∨ ¬ ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) ) ) )
35 32 34 cnf1dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ( 𝑥𝐴 ∨ ¬ 𝑥𝐵 ) → ¬ ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) ) )
36 30 35 contrd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( 𝑥𝐴 ∨ ¬ 𝑥𝐵 ) )
37 36 ord ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ 𝑥𝐴 → ¬ 𝑥𝐵 ) )
38 tsan2 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( 𝑥𝐵 ∨ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) ) )
39 38 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ 𝑥𝐴 → ( 𝑥𝐵 ∨ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) )
40 37 39 cnf1dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ 𝑥𝐴 → ¬ ( 𝑥𝐵𝑦 = 𝐸 ) ) )
41 28 40 contrd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → 𝑥𝐴 )
42 41 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → 𝑥𝐴 ) )
43 29 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) ) )
44 tsan3 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ∨ ¬ ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) ) )
45 44 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ∨ ¬ ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) ) ) )
46 43 45 cnfn2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) )
47 42 46 mpdd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) )
48 notnotr ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ( 𝑥𝐵𝑦 = 𝐸 ) )
49 48 a1i ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ( 𝑥𝐵𝑦 = 𝐸 ) ) )
50 38 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ( 𝑥𝐵 ∨ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) )
51 49 50 cnfn2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → 𝑥𝐵 ) )
52 36 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ( 𝑥𝐴 ∨ ¬ 𝑥𝐵 ) ) )
53 51 52 cnfn2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → 𝑥𝐴 ) )
54 tsan3 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( 𝑦 = 𝐸 ∨ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) ) )
55 54 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ( 𝑦 = 𝐸 ∨ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) )
56 49 55 cnfn2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → 𝑦 = 𝐸 ) )
57 29 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) ) )
58 44 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ( ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ∨ ¬ ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) ) ) )
59 57 58 cnfn2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) )
60 53 59 mpdd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) )
61 tsbi3 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ( 𝑦 = 𝐷 ∨ ¬ 𝑦 = 𝐸 ) ∨ ¬ ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) )
62 61 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ( ( 𝑦 = 𝐷 ∨ ¬ 𝑦 = 𝐸 ) ∨ ¬ ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) )
63 60 62 cnfn2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ( 𝑦 = 𝐷 ∨ ¬ 𝑦 = 𝐸 ) ) )
64 56 63 cnfn2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → 𝑦 = 𝐷 ) )
65 53 64 jcad ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ( 𝑥𝐴𝑦 = 𝐷 ) ) )
66 ax-1 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) ) )
67 tsim3 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ∨ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) ) )
68 67 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ( ¬ ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ∨ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) ) ) )
69 66 68 cnf2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ¬ ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) )
70 tsbi1 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ( ¬ ( 𝑥𝐴𝑦 = 𝐷 ) ∨ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) ) ∨ ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) )
71 70 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ( ( ¬ ( 𝑥𝐴𝑦 = 𝐷 ) ∨ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) ) ∨ ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) ) )
72 69 71 cnf2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ( ¬ ( 𝑥𝐴𝑦 = 𝐷 ) ∨ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) )
73 49 72 cnfn2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥𝐵𝑦 = 𝐸 ) → ¬ ( 𝑥𝐴𝑦 = 𝐷 ) ) )
74 65 73 contrd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ¬ ( 𝑥𝐵𝑦 = 𝐸 ) )
75 74 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ¬ ( 𝑥𝐵𝑦 = 𝐸 ) ) )
76 26 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( 𝑥𝐴𝑦 = 𝐷 ) ∨ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) )
77 75 76 cnf2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( 𝑥𝐴𝑦 = 𝐷 ) ) )
78 tsan3 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( 𝑦 = 𝐷 ∨ ¬ ( 𝑥𝐴𝑦 = 𝐷 ) ) )
79 78 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( 𝑦 = 𝐷 ∨ ¬ ( 𝑥𝐴𝑦 = 𝐷 ) ) ) )
80 77 79 cnfn2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → 𝑦 = 𝐷 ) )
81 33 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( 𝑥𝐴𝑥𝐵 ) ∨ ¬ ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) ) ) )
82 43 81 cnfn2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( 𝑥𝐴𝑥𝐵 ) ) )
83 tsbi4 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ( ¬ 𝑥𝐴𝑥𝐵 ) ∨ ¬ ( 𝑥𝐴𝑥𝐵 ) ) )
84 83 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( ¬ 𝑥𝐴𝑥𝐵 ) ∨ ¬ ( 𝑥𝐴𝑥𝐵 ) ) ) )
85 82 84 cnfn2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ¬ 𝑥𝐴𝑥𝐵 ) ) )
86 42 85 cnfn1dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → 𝑥𝐵 ) )
87 tsan1 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ( ¬ 𝑥𝐵 ∨ ¬ 𝑦 = 𝐸 ) ∨ ( 𝑥𝐵𝑦 = 𝐸 ) ) )
88 87 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( ¬ 𝑥𝐵 ∨ ¬ 𝑦 = 𝐸 ) ∨ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) )
89 75 88 cnf2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ¬ 𝑥𝐵 ∨ ¬ 𝑦 = 𝐸 ) ) )
90 86 89 cnfn1dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ¬ 𝑦 = 𝐸 ) )
91 tsbi4 ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ( ¬ 𝑦 = 𝐷𝑦 = 𝐸 ) ∨ ¬ ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) )
92 91 a1d ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( ¬ 𝑦 = 𝐷𝑦 = 𝐸 ) ∨ ¬ ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) )
93 92 or32dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( ¬ 𝑦 = 𝐷 ∨ ¬ ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ∨ 𝑦 = 𝐸 ) ) )
94 90 93 cnf2dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ¬ 𝑦 = 𝐷 ∨ ¬ ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) )
95 80 94 cnfn1dd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ¬ ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) )
96 47 95 contrd ( ¬ ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) ) → ⊥ )
97 96 efald2 ( ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) )
98 97 2alimi ( ∀ 𝑥𝑦 ( ( 𝑥𝐴𝑥𝐵 ) ∧ ( 𝑥𝐴 → ( 𝑦 = 𝐷𝑦 = 𝐸 ) ) ) → ∀ 𝑥𝑦 ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) )
99 19 98 syl ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥𝐴 𝐷 = 𝐸 ) → ∀ 𝑥𝑦 ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) )
100 eqopab2bw ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐴𝑦 = 𝐷 ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐵𝑦 = 𝐸 ) } ↔ ∀ 𝑥𝑦 ( ( 𝑥𝐴𝑦 = 𝐷 ) ↔ ( 𝑥𝐵𝑦 = 𝐸 ) ) )
101 99 100 sylibr ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥𝐴 𝐷 = 𝐸 ) → { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐴𝑦 = 𝐷 ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐵𝑦 = 𝐸 ) } )
102 df-mpt ( 𝑥𝐴𝐷 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐴𝑦 = 𝐷 ) }
103 df-mpt ( 𝑥𝐵𝐸 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐵𝑦 = 𝐸 ) }
104 101 102 103 3eqtr4g ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥𝐴 𝐷 = 𝐸 ) → ( 𝑥𝐴𝐷 ) = ( 𝑥𝐵𝐸 ) )