Theorem cbvoprab3v 5954

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 1528	. 2 |- F/ w ph
2		nfv 1528	. 2 |- F/ z ps
3		cbvoprab3v.1	. 2 |- ( z = w -> ( ph <-> ps ) )
4	1, 2, 3	cbvoprab3 5953	1 |- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , w >. | ps }

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   {coprab 5878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-opab 4067  df-oprab 5881

This theorem is referenced by: (None)