Theorem cbvoprab12v 6730

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 1843	. 2 |- F/ w ph
2		nfv 1843	. 2 |- F/ v ph
3		nfv 1843	. 2 |- F/ x ps
4		nfv 1843	. 2 |- F/ y ps
5		cbvoprab12v.1	. 2 |- ( ( x = w /\ y = v ) -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvoprab12 6729	1 |- { <. <. x , y >. , z >. | ph } = { <. <. w , v >. , z >. | ps }

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   {coprab 6651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-oprab 6654

This theorem is referenced by:  cpnnen  14958