Theorem cbvopab1v 4012

Description: Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Hypothesis

Ref	Expression
cbvopab1v.1	|- ( x = z -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvopab1v	|- { <. x , y >. | ph } = { <. z , y >. | ps }

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

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

Proof of Theorem cbvopab1v

Step	Hyp	Ref	Expression
1		nfv 1509	. 2 |- F/ z ph
2		nfv 1509	. 2 |- F/ x ps
3		cbvopab1v.1	. 2 |- ( x = z -> ( ph <-> ps ) )
4	1, 2, 3	cbvopab1 4009	1 |- { <. x , y >. | ph } = { <. z , y >. | ps }

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   {copab 3996

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-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

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-sn 3538  df-pr 3539  df-op 3541  df-opab 3998

This theorem is referenced by: (None)