Theorem cbvrabv 2858

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)

Hypothesis

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

Assertion

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

Distinct variable groups:   x,y,A   φ,y   ψ,x

Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem cbvrabv

Step	Hyp	Ref	Expression
1		nfcv 2489	. 2 ⊢ ℲxA
2		nfcv 2489	. 2 ⊢ ℲyA
3		nfv 1619	. 2 ⊢ Ⅎyφ
4		nfv 1619	. 2 ⊢ Ⅎxψ
5		cbvrabv.1	. 2 ⊢ (x = y → (φ ↔ ψ))
6	1, 2, 3, 4, 5	cbvrab 2857	1 ⊢ {x ∈ A ∣ φ} = {y ∈ A ∣ ψ}

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  {crab 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rab 2623

This theorem is referenced by: (None)