Theorem cbvralv2 3147

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 2336	. 2 |- F/_ y A
2		nfcv 2336	. 2 |- F/_ x B
3		nfv 1539	. 2 |- F/ y ps
4		nfv 1539	. 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 3143	1 |- ( A. x e. A ps <-> A. y e. B ch )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   A.wral 2472

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-sbc 2986  df-csb 3081

This theorem is referenced by: (None)