Metamath Proof Explorer


Theorem bnj1533

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1533.1 ( 𝜃 → ∀ 𝑧𝐵 ¬ 𝑧𝐷 )
bnj1533.2 𝐵𝐴
bnj1533.3 𝐷 = { 𝑧𝐴𝐶𝐸 }
Assertion bnj1533 ( 𝜃 → ∀ 𝑧𝐵 𝐶 = 𝐸 )

Proof

Step Hyp Ref Expression
1 bnj1533.1 ( 𝜃 → ∀ 𝑧𝐵 ¬ 𝑧𝐷 )
2 bnj1533.2 𝐵𝐴
3 bnj1533.3 𝐷 = { 𝑧𝐴𝐶𝐸 }
4 1 bnj1211 ( 𝜃 → ∀ 𝑧 ( 𝑧𝐵 → ¬ 𝑧𝐷 ) )
5 3 rabeq2i ( 𝑧𝐷 ↔ ( 𝑧𝐴𝐶𝐸 ) )
6 5 notbii ( ¬ 𝑧𝐷 ↔ ¬ ( 𝑧𝐴𝐶𝐸 ) )
7 imnan ( ( 𝑧𝐴 → ¬ 𝐶𝐸 ) ↔ ¬ ( 𝑧𝐴𝐶𝐸 ) )
8 nne ( ¬ 𝐶𝐸𝐶 = 𝐸 )
9 8 imbi2i ( ( 𝑧𝐴 → ¬ 𝐶𝐸 ) ↔ ( 𝑧𝐴𝐶 = 𝐸 ) )
10 6 7 9 3bitr2i ( ¬ 𝑧𝐷 ↔ ( 𝑧𝐴𝐶 = 𝐸 ) )
11 10 imbi2i ( ( 𝑧𝐵 → ¬ 𝑧𝐷 ) ↔ ( 𝑧𝐵 → ( 𝑧𝐴𝐶 = 𝐸 ) ) )
12 2 sseli ( 𝑧𝐵𝑧𝐴 )
13 12 imim1i ( ( 𝑧𝐴𝐶 = 𝐸 ) → ( 𝑧𝐵𝐶 = 𝐸 ) )
14 ax-1 ( ( 𝑧𝐴𝐶 = 𝐸 ) → ( 𝑧𝐵 → ( 𝑧𝐴𝐶 = 𝐸 ) ) )
15 14 anim1i ( ( ( 𝑧𝐴𝐶 = 𝐸 ) ∧ 𝑧𝐵 ) → ( ( 𝑧𝐵 → ( 𝑧𝐴𝐶 = 𝐸 ) ) ∧ 𝑧𝐵 ) )
16 simpr ( ( ( 𝑧𝐵 → ( 𝑧𝐴𝐶 = 𝐸 ) ) ∧ 𝑧𝐵 ) → 𝑧𝐵 )
17 simpl ( ( ( 𝑧𝐵 → ( 𝑧𝐴𝐶 = 𝐸 ) ) ∧ 𝑧𝐵 ) → ( 𝑧𝐵 → ( 𝑧𝐴𝐶 = 𝐸 ) ) )
18 16 17 mpd ( ( ( 𝑧𝐵 → ( 𝑧𝐴𝐶 = 𝐸 ) ) ∧ 𝑧𝐵 ) → ( 𝑧𝐴𝐶 = 𝐸 ) )
19 18 16 jca ( ( ( 𝑧𝐵 → ( 𝑧𝐴𝐶 = 𝐸 ) ) ∧ 𝑧𝐵 ) → ( ( 𝑧𝐴𝐶 = 𝐸 ) ∧ 𝑧𝐵 ) )
20 15 19 impbii ( ( ( 𝑧𝐴𝐶 = 𝐸 ) ∧ 𝑧𝐵 ) ↔ ( ( 𝑧𝐵 → ( 𝑧𝐴𝐶 = 𝐸 ) ) ∧ 𝑧𝐵 ) )
21 20 imbi1i ( ( ( ( 𝑧𝐴𝐶 = 𝐸 ) ∧ 𝑧𝐵 ) → 𝐶 = 𝐸 ) ↔ ( ( ( 𝑧𝐵 → ( 𝑧𝐴𝐶 = 𝐸 ) ) ∧ 𝑧𝐵 ) → 𝐶 = 𝐸 ) )
22 impexp ( ( ( ( 𝑧𝐴𝐶 = 𝐸 ) ∧ 𝑧𝐵 ) → 𝐶 = 𝐸 ) ↔ ( ( 𝑧𝐴𝐶 = 𝐸 ) → ( 𝑧𝐵𝐶 = 𝐸 ) ) )
23 impexp ( ( ( ( 𝑧𝐵 → ( 𝑧𝐴𝐶 = 𝐸 ) ) ∧ 𝑧𝐵 ) → 𝐶 = 𝐸 ) ↔ ( ( 𝑧𝐵 → ( 𝑧𝐴𝐶 = 𝐸 ) ) → ( 𝑧𝐵𝐶 = 𝐸 ) ) )
24 21 22 23 3bitr3i ( ( ( 𝑧𝐴𝐶 = 𝐸 ) → ( 𝑧𝐵𝐶 = 𝐸 ) ) ↔ ( ( 𝑧𝐵 → ( 𝑧𝐴𝐶 = 𝐸 ) ) → ( 𝑧𝐵𝐶 = 𝐸 ) ) )
25 13 24 mpbi ( ( 𝑧𝐵 → ( 𝑧𝐴𝐶 = 𝐸 ) ) → ( 𝑧𝐵𝐶 = 𝐸 ) )
26 11 25 sylbi ( ( 𝑧𝐵 → ¬ 𝑧𝐷 ) → ( 𝑧𝐵𝐶 = 𝐸 ) )
27 4 26 sylg ( 𝜃 → ∀ 𝑧 ( 𝑧𝐵𝐶 = 𝐸 ) )
28 27 bnj1142 ( 𝜃 → ∀ 𝑧𝐵 𝐶 = 𝐸 )