Theorem cbvopab1v 3999

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 1508	. 2 |- F/ z ph
2		nfv 1508	. 2 |- F/ x ps
3		cbvopab1v.1	. 2 |- ( x = z -> ( ph <-> ps ) )
4	1, 2, 3	cbvopab1 3996	1 |- { <. x , y >. | ph } = { <. z , y >. | ps }

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   {copab 3983

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-opab 3985

This theorem is referenced by: (None)