Theorem cbvoprab12v 6043

Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.)

Hypothesis

Ref	Expression
cbvoprab12v.1	|- ( ( x = w /\ y = v ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvoprab12v	|- { <. <. x , y >. , z >. | ph } = { <. <. w , v >. , z >. | ps }

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

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

Proof of Theorem cbvoprab12v

Step	Hyp	Ref	Expression
1		nfv 1552	. 2 |- F/ w ph
2		nfv 1552	. 2 |- F/ v ph
3		nfv 1552	. 2 |- F/ x ps
4		nfv 1552	. 2 |- F/ y ps
5		cbvoprab12v.1	. 2 |- ( ( x = w /\ y = v ) -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvoprab12 6042	1 |- { <. <. x , y >. , z >. | ph } = { <. <. w , v >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   {coprab 5968

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-opab 4122  df-oprab 5971

This theorem is referenced by: (None)