Theorem cbvopabv 3829

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 1421	. 2 |- F/ z ph
2		nfv 1421	. 2 |- F/ w ph
3		nfv 1421	. 2 |- F/ x ps
4		nfv 1421	. 2 |- F/ y ps
5		cbvopabv.1	. 2 |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvopab 3828	1 |- { <. x , y >. | ph } = { <. z , w >. | ps }

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   {copab 3817

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819

This theorem is referenced by: (None)