Theorem cbvcllem 37915

Description: Change of bound variable in class of supersets of a with a property. (Contributed by RP, 24-Jul-2020.)

Hypothesis

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

Assertion

Ref	Expression
cbvcllem	|- { x | ( X C_ x /\ ph ) } = { y | ( X C_ y /\ ps ) }

Distinct variable groups:   x, y, X   ps, x   ph, y

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

Proof of Theorem cbvcllem

Step	Hyp	Ref	Expression
1		cbvcllem.y	. . 3 |- ( x = y -> ( ph <-> ps ) )
2	1	cleq2lem 37914	. 2 |- ( x = y -> ( ( X C_ x /\ ph ) <-> ( X C_ y /\ ps ) ) )
3	2	cbvabv 2747	1 |- { x | ( X C_ x /\ ph ) } = { y | ( X C_ y /\ ps ) }

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   {cab 2608   C_ wss 3574

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-in 3581  df-ss 3588

This theorem is referenced by: (None)