Metamath Proof Explorer

Theorem cgsex4g

Description: An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995) Avoid ax-10 , ax-11 . (Revised by Gino Giotto, 28-Jun-2024)

Ref Expression
Hypotheses cgsex4g.1 ⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → 𝜒 )
cgsex4g.2 ⊢ ( 𝜒 → ( 𝜑 ↔ 𝜓 ) )
Assertion cgsex4g ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ↔ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 cgsex4g.1 ⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → 𝜒 )
2 cgsex4g.2 ⊢ ( 𝜒 → ( 𝜑 ↔ 𝜓 ) )
3 2 biimpa ⊢ ( ( 𝜒 ∧ 𝜑 ) → 𝜓 )
4 3 exlimivv ⊢ ( ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) → 𝜓 )
5 4 exlimivv ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) → 𝜓 )
6 elisset ⊢ ( 𝐴 ∈ 𝑅 → ∃ 𝑥 𝑥 = 𝐴 )
7 elisset ⊢ ( 𝐵 ∈ 𝑆 → ∃ 𝑦 𝑦 = 𝐵 )
8 6 7 anim12i ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
9 exdistrv ⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
10 8 9 sylibr ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) )
11 elisset ⊢ ( 𝐶 ∈ 𝑅 → ∃ 𝑧 𝑧 = 𝐶 )
12 elisset ⊢ ( 𝐷 ∈ 𝑆 → ∃ 𝑤 𝑤 = 𝐷 )
13 11 12 anim12i ⊢ ( ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) → ( ∃ 𝑧 𝑧 = 𝐶 ∧ ∃ 𝑤 𝑤 = 𝐷 ) )
14 exdistrv ⊢ ( ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ↔ ( ∃ 𝑧 𝑧 = 𝐶 ∧ ∃ 𝑤 𝑤 = 𝐷 ) )
15 13 14 sylibr ⊢ ( ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) → ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) )
16 10 15 anim12i ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
17 eqeq1 ⊢ ( 𝑧 = 𝑣 → ( 𝑧 = 𝐶 ↔ 𝑣 = 𝐶 ) )
18 17 anbi1d ⊢ ( 𝑧 = 𝑣 → ( ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ↔ ( 𝑣 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
19 18 anbi2d ⊢ ( 𝑧 = 𝑣 → ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑣 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) )
20 19 exbidv ⊢ ( 𝑧 = 𝑣 → ( ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑣 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) )
21 20 notbid ⊢ ( 𝑧 = 𝑣 → ( ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑣 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) )
22 21 alcomiw ⊢ ( ∀ 𝑦 ∀ 𝑧 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → ∀ 𝑧 ∀ 𝑦 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
23 eqeq1 ⊢ ( 𝑦 = 𝑣 → ( 𝑦 = 𝐵 ↔ 𝑣 = 𝐵 ) )
24 23 anbi2d ⊢ ( 𝑦 = 𝑣 → ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( 𝑥 = 𝐴 ∧ 𝑣 = 𝐵 ) ) )
25 24 anbi1d ⊢ ( 𝑦 = 𝑣 → ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ( ( 𝑥 = 𝐴 ∧ 𝑣 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) )
26 25 exbidv ⊢ ( 𝑦 = 𝑣 → ( ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑣 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) )
27 26 notbid ⊢ ( 𝑦 = 𝑣 → ( ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑣 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) )
28 27 alcomiw ⊢ ( ∀ 𝑧 ∀ 𝑦 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → ∀ 𝑦 ∀ 𝑧 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
29 22 28 impbii ⊢ ( ∀ 𝑦 ∀ 𝑧 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ∀ 𝑧 ∀ 𝑦 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
30 29 notbii ⊢ ( ¬ ∀ 𝑦 ∀ 𝑧 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ¬ ∀ 𝑧 ∀ 𝑦 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
31 2exnaln ⊢ ( ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ¬ ∀ 𝑦 ∀ 𝑧 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
32 2exnaln ⊢ ( ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ¬ ∀ 𝑧 ∀ 𝑦 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
33 30 31 32 3bitr4i ⊢ ( ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
34 33 exbii ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ∃ 𝑥 ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
35 4exdistrv ⊢ ( ∃ 𝑥 ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
36 34 35 bitri ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
37 16 36 sylibr ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
38 1 2eximi ⊢ ( ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → ∃ 𝑧 ∃ 𝑤 𝜒 )
39 38 2eximi ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 𝜒 )
40 37 39 syl ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 𝜒 )
41 2 biimprcd ⊢ ( 𝜓 → ( 𝜒 → 𝜑 ) )
42 41 ancld ⊢ ( 𝜓 → ( 𝜒 → ( 𝜒 ∧ 𝜑 ) ) )
43 42 2eximdv ⊢ ( 𝜓 → ( ∃ 𝑧 ∃ 𝑤 𝜒 → ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ) )
44 43 2eximdv ⊢ ( 𝜓 → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 𝜒 → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ) )
45 40 44 syl5com ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( 𝜓 → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ) )
46 5 45 impbid2 ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ↔ 𝜓 ) )