Theorem cbvopab1v 4170

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 1577	. 2 |- F/ z ph
2		nfv 1577	. 2 |- F/ x ps
3		cbvopab1v.1	. 2 |- ( x = z -> ( ph <-> ps ) )
4	1, 2, 3	cbvopab1 4167	1 |- { <. x , y >. | ph } = { <. z , y >. | ps }

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   {copab 4154

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156

This theorem is referenced by: (None)