Theorem cbvopabv 3960

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 1491	. 2 |- F/ z ph
2		nfv 1491	. 2 |- F/ w ph
3		nfv 1491	. 2 |- F/ x ps
4		nfv 1491	. 2 |- F/ y ps
5		cbvopabv.1	. 2 |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvopab 3959	1 |- { <. x , y >. | ph } = { <. z , w >. | ps }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1314   {copab 3948

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500  df-op 3502  df-opab 3950

This theorem is referenced by: (None)