Theorem cbvrabv 1426

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution.

Hypothesis

Ref	Expression
cbvrabv.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvrabv	⊢ {x ∈ A∣φ} = {y ∈ A∣ψ}

Distinct variable group(s):   x,y,A   φ,y   ψ,x

Proof of Theorem cbvrabv

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (z ∈ A → ∀x z ∈ A)
2		ax-17 925	. 2 ⊢ (z ∈ A → ∀y z ∈ A)
3		ax-17 925	. 2 ⊢ (φ → ∀yφ)
4		ax-17 925	. 2 ⊢ (ψ → ∀xψ)
5		cbvrabv.1	. 2 ⊢ (x = y → (φ ↔ ψ))
6	1, 2, 3, 4, 5	cbvrab 1425	1 ⊢ {x ∈ A∣φ} = {y ∈ A∣ψ}

Syntax hints:   → wi 2   ↔ wb 127   = weq 797   = wceq 1091   ∈ wcel 1092  {crab 1204

This theorem is referenced by:  reuuni3 1958  inf3lema 3460  zorn 3611  uzwo3lem2 4615  sqrlem24 4754  sqrgt0i 4755  sqrlem26 4756

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-rab 1208  df-v 1349

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