Theorem cbvoprab12v 5854

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 1509	. 2 |- F/ w ph
2		nfv 1509	. 2 |- F/ v ph
3		nfv 1509	. 2 |- F/ x ps
4		nfv 1509	. 2 |- F/ y ps
5		cbvoprab12v.1	. 2 |- ( ( x = w /\ y = v ) -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvoprab12 5853	1 |- { <. <. x , y >. , z >. | ph } = { <. <. w , v >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   {coprab 5783

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-opab 3998  df-oprab 5786

This theorem is referenced by: (None)