Theorem cbvoprab3v 5601

Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) (Revised by David Abernethy, 19-Jun-2012.)

Hypothesis

Ref	Expression
cbvoprab3v.1	|- ( z = w -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvoprab3v	|- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , w >. | ps }

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

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

Proof of Theorem cbvoprab3v

Step	Hyp	Ref	Expression
1		nfv 1461	. 2 |- F/ w ph
2		nfv 1461	. 2 |- F/ z ps
3		cbvoprab3v.1	. 2 |- ( z = w -> ( ph <-> ps ) )
4	1, 2, 3	cbvoprab3 5600	1 |- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , w >. | ps }

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   {coprab 5533

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840  df-oprab 5536

This theorem is referenced by: (None)