Theorem cbvoprab12v 5579

Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.)

Hypothesis

Ref	Expression
cbvoprab12v.1	|- ( ( x = w /\ y = v ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvoprab12v	|- { <. <. x , y >. , z >. | ph } = { <. <. w , v >. , z >. | ps }

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

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

Proof of Theorem cbvoprab12v

Step	Hyp	Ref	Expression
1		nfv 1421	. 2 |- F/ w ph
2		nfv 1421	. 2 |- F/ v ph
3		nfv 1421	. 2 |- F/ x ps
4		nfv 1421	. 2 |- F/ y ps
5		cbvoprab12v.1	. 2 |- ( ( x = w /\ y = v ) -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvoprab12 5578	1 |- { <. <. x , y >. , z >. | ph } = { <. <. w , v >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   {coprab 5513

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-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

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-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-oprab 5516

This theorem is referenced by: (None)