Theorem cbvoprab3v 6022

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 1551	. 2 |- F/ w ph
2		nfv 1551	. 2 |- F/ z ps
3		cbvoprab3v.1	. 2 |- ( z = w -> ( ph <-> ps ) )
4	1, 2, 3	cbvoprab3 6021	1 |- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , w >. | ps }

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   {coprab 5945

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-opab 4106  df-oprab 5948

This theorem is referenced by: (None)