Theorem cbvopabv 4048

Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)

Hypothesis

Ref	Expression
cbvopabv.1	|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvopabv	|- { <. x , y >. | ph } = { <. z , w >. | ps }

Distinct variable groups:   x, y, z, w   ph, z, w   ps, x, y

Allowed substitution hints:   ph( x, y)   ps( z, w)

Proof of Theorem cbvopabv

Step	Hyp	Ref	Expression
1		nfv 1515	. 2 |- F/ z ph
2		nfv 1515	. 2 |- F/ w ph
3		nfv 1515	. 2 |- F/ x ps
4		nfv 1515	. 2 |- F/ y ps
5		cbvopabv.1	. 2 |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvopab 4047	1 |- { <. x , y >. | ph } = { <. z , w >. | ps }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1342   {copab 4036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115  df-sn 3576  df-pr 3577  df-op 3579  df-opab 4038

This theorem is referenced by: (None)