Theorem ovmpt2dv 6793

Description: Alternate deduction version of ovmpt2 6796, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Hypotheses

Ref	Expression
ovmpt2df.1	|- ( ph -> A e. C )
ovmpt2df.2	|- ( ( ph /\ x = A ) -> B e. D )
ovmpt2df.3	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
ovmpt2df.4	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = R -> ps ) )

Assertion

Ref	Expression
ovmpt2dv	|- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )

Distinct variable groups:   x, y, A   y, B   ph, x, y   x, F, y   ps, x, y

Allowed substitution hints:   B( x)   C( x, y)   D( x, y)   R( x, y)   V( x, y)

Proof of Theorem ovmpt2dv

Step	Hyp	Ref	Expression
1		ovmpt2df.1	. 2 |- ( ph -> A e. C )
2		ovmpt2df.2	. 2 |- ( ( ph /\ x = A ) -> B e. D )
3		ovmpt2df.3	. 2 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
4		ovmpt2df.4	. 2 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = R -> ps ) )
5		nfcv 2764	. 2 |- F/_ x F
6		nfv 1843	. 2 |- F/ x ps
7		nfcv 2764	. 2 |- F/_ y F
8		nfv 1843	. 2 |- F/ y ps
9	1, 2, 3, 4, 5, 6, 7, 8	ovmpt2df 6792	1 |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990  (class class class)co 6650   |-> cmpt2 6652

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  xpcco  16823  curf12  16867  curf2  16869