Theorem cbvopab1v 3856

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 1462	. 2 |- F/ z ph
2		nfv 1462	. 2 |- F/ x ps
3		cbvopab1v.1	. 2 |- ( x = z -> ( ph <-> ps ) )
4	1, 2, 3	cbvopab1 3853	1 |- { <. x , y >. | ph } = { <. z , y >. | ps }

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1285   {copab 3840

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-opab 3842

This theorem is referenced by: (None)