Theorem cbvsbcv 3465

Description: Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypothesis

Ref	Expression
cbvsbcv.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvsbcv	|- ( [. A / x ]. ph <-> [. A / y ]. ps )

Distinct variable groups:   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)   A( x, y)

Proof of Theorem cbvsbcv

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ y ph
2		nfv 1843	. 2 |- F/ x ps
3		cbvsbcv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvsbc 3464	1 |- ( [. A / x ]. ph <-> [. A / y ]. ps )

Syntax hints:   -> wi 4   <-> wb 196   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by:  fpwwe2cbv  9452