Theorem cbvralv2 3124

Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
cbvralv2.1	|- ( x = y -> ( ps <-> ch ) )
cbvralv2.2	|- ( x = y -> A = B )

Assertion

Ref	Expression
cbvralv2	|- ( A. x e. A ps <-> A. y e. B ch )

Distinct variable groups:   y, A   ps, y   x, B   ch, x

Allowed substitution hints:   ps( x)   ch( y)   A( x)   B( y)

Proof of Theorem cbvralv2

Step	Hyp	Ref	Expression
1		nfcv 2319	. 2 |- F/_ y A
2		nfcv 2319	. 2 |- F/_ x B
3		nfv 1528	. 2 |- F/ y ps
4		nfv 1528	. 2 |- F/ x ch
5		cbvralv2.2	. 2 |- ( x = y -> A = B )
6		cbvralv2.1	. 2 |- ( x = y -> ( ps <-> ch ) )
7	1, 2, 3, 4, 5, 6	cbvralcsf 3120	1 |- ( A. x e. A ps <-> A. y e. B ch )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   A.wral 2455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-sbc 2964  df-csb 3059

This theorem is referenced by: (None)