Theorem ovmpodv2 6165

Description: Alternate deduction version of ovmpo 6167, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Hypotheses

Ref	Expression
ovmpodv2.1	|- ( ph -> A e. C )
ovmpodv2.2	|- ( ( ph /\ x = A ) -> B e. D )
ovmpodv2.3	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
ovmpodv2.4	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )

Assertion

Ref	Expression
ovmpodv2	|- ( 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 ovmpodv2

Step	Hyp	Ref	Expression
1		eqidd 2232	. . 3 |- ( ph -> ( x e. C , y e. D |-> R ) = ( x e. C , y e. D |-> R ) )
2		ovmpodv2.1	. . . 4 |- ( ph -> A e. C )
3		ovmpodv2.2	. . . 4 |- ( ( ph /\ x = A ) -> B e. D )
4		ovmpodv2.3	. . . 4 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
5		ovmpodv2.4	. . . . . 6 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
6	5	eqeq2d 2243	. . . . 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 144	. . . 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		nfmpo1 6098	. . . 4 |- F/_ x ( x e. C , y e. D |-> R )
9		nfcv 2375	. . . . . 6 |- F/_ x A
10		nfcv 2375	. . . . . 6 |- F/_ x B
11	9, 8, 10	nfov 6058	. . . . 5 |- F/_ x ( A ( x e. C , y e. D |-> R ) B )
12	11	nfeq1 2385	. . . 4 |- F/ x ( A ( x e. C , y e. D |-> R ) B ) = S
13		nfmpo2 6099	. . . 4 |- F/_ y ( x e. C , y e. D |-> R )
14		nfcv 2375	. . . . . 6 |- F/_ y A
15		nfcv 2375	. . . . . 6 |- F/_ y B
16	14, 13, 15	nfov 6058	. . . . 5 |- F/_ y ( A ( x e. C , y e. D |-> R ) B )
17	16	nfeq1 2385	. . . 4 |- F/ y ( A ( x e. C , y e. D |-> R ) B ) = S
18	2, 3, 4, 7, 8, 12, 13, 17	ovmpodf 6163	. . 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 13	. 2 |- ( ph -> ( A ( x e. C , y e. D |-> R ) B ) = S )
20		oveq 6034	. . 3 |- ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) )
21	20	eqeq1d 2240	. 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 157	1 |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202  (class class class)co 6028   e. cmpo 6030

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-in1 619  ax-in2 620  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-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033

This theorem is referenced by: (None)