Theorem ovmpt2dv2 6794

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

Hypotheses

Ref	Expression
ovmpt2dv2.1	|- ( ph -> A e. C )
ovmpt2dv2.2	|- ( ( ph /\ x = A ) -> B e. D )
ovmpt2dv2.3	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
ovmpt2dv2.4	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )

Assertion

Ref	Expression
ovmpt2dv2	|- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = S ) )

Distinct variable groups:   x, y, A   x, B, y   ph, x, y   x, S, y

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

Proof of Theorem ovmpt2dv2

Step	Hyp	Ref	Expression
1		eqidd 2623	. . 3 |- ( ph -> ( x e. C , y e. D |-> R ) = ( x e. C , y e. D |-> R ) )
2		ovmpt2dv2.1	. . . 4 |- ( ph -> A e. C )
3		ovmpt2dv2.2	. . . 4 |- ( ( ph /\ x = A ) -> B e. D )
4		ovmpt2dv2.3	. . . 4 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
5		ovmpt2dv2.4	. . . . . 6 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
6	5	eqeq2d 2632	. . . . 5 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A ( x e. C , y e. D |-> R ) B ) = R <-> ( A ( x e. C , y e. D |-> R ) B ) = S ) )
7	6	biimpd 219	. . . 4 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A ( x e. C , y e. D |-> R ) B ) = R -> ( A ( x e. C , y e. D |-> R ) B ) = S ) )
8		nfmpt21 6722	. . . 4 |- F/_ x ( x e. C , y e. D |-> R )
9		nfcv 2764	. . . . . 6 |- F/_ x A
10		nfcv 2764	. . . . . 6 |- F/_ x B
11	9, 8, 10	nfov 6676	. . . . 5 |- F/_ x ( A ( x e. C , y e. D |-> R ) B )
12	11	nfeq1 2778	. . . 4 |- F/ x ( A ( x e. C , y e. D |-> R ) B ) = S
13		nfmpt22 6723	. . . 4 |- F/_ y ( x e. C , y e. D |-> R )
14		nfcv 2764	. . . . . 6 |- F/_ y A
15		nfcv 2764	. . . . . 6 |- F/_ y B
16	14, 13, 15	nfov 6676	. . . . 5 |- F/_ y ( A ( x e. C , y e. D |-> R ) B )
17	16	nfeq1 2778	. . . 4 |- F/ y ( A ( x e. C , y e. D |-> R ) B ) = S
18	2, 3, 4, 7, 8, 12, 13, 17	ovmpt2df 6792	. . 3 |- ( ph -> ( ( x e. C , y e. D |-> R ) = ( x e. C , y e. D |-> R ) -> ( A ( x e. C , y e. D |-> R ) B ) = S ) )
19	1, 18	mpd 15	. 2 |- ( ph -> ( A ( x e. C , y e. D |-> R ) B ) = S )
20		oveq 6656	. . 3 |- ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) )
21	20	eqeq1d 2624	. 2 |- ( F = ( x e. C , y e. D |-> R ) -> ( ( A F B ) = S <-> ( A ( x e. C , y e. D |-> R ) B ) = S ) )
22	19, 21	syl5ibrcom 237	1 |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = S ) )

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:  coaval  16718  xpcco  16823  marrepval  20368  marrepeval  20369  marepveval  20374  submaval  20387  submaeval  20388  minmar1val  20454  minmar1eval  20455  nbgrval  26234