Theorem cbvab 1423

Description: Rule used to change bound variables with implicit substitution.

Hypotheses

Ref	Expression
cbvab.1	⊢ (φ → ∀yφ)
cbvab.2	⊢ (ψ → ∀xψ)
cbvab.3	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvab	⊢ {x∣φ} = {y∣ψ}

Distinct variable group(s):   x,y

Proof of Theorem cbvab

Step	Hyp	Ref	Expression
1		cbvab.1	. . . 4 ⊢ (φ → ∀yφ)
2	1	hbab 1096	. . 3 ⊢ (z ∈ {x∣φ} → ∀y z ∈ {x∣φ})
3		hbab1 1095	. . 3 ⊢ (z ∈ {y∣ψ} → ∀y z ∈ {y∣ψ})
4	2, 3	cleqf 1167	. 2 ⊢ ({x∣φ} = {y∣ψ} ↔ ∀y(y ∈ {x∣φ} ↔ y ∈ {y∣ψ}))
5		cbvab.2	. . . 4 ⊢ (ψ → ∀xψ)
6		visset 1350	. . . 4 ⊢ y ∈ V
7		cbvab.3	. . . 4 ⊢ (x = y → (φ ↔ ψ))
8	5, 6, 7	elabf 1414	. . 3 ⊢ (y ∈ {x∣φ} ↔ ψ)
9		abid 1094	. . 3 ⊢ (y ∈ {y∣ψ} ↔ ψ)
10	8, 9	bitr4 154	. 2 ⊢ (y ∈ {x∣φ} ↔ y ∈ {y∣ψ})
11	4, 10	mpgbir 686	1 ⊢ {x∣φ} = {y∣ψ}

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  cbvabv 1424  cbvrab 1425  dfdmf 2526  dfrnf 2561  abrexexlem2 2911  abrexex2 2915

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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