Theorem ovmpodv2 5986

Description: Alternate deduction version of ovmpo 5988, 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 2171	. . 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 2182	. . . . 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 143	. . . 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 5920	. . . 4 |- F/_ x ( x e. C , y e. D |-> R )
9		nfcv 2312	. . . . . 6 |- F/_ x A
10		nfcv 2312	. . . . . 6 |- F/_ x B
11	9, 8, 10	nfov 5883	. . . . 5 |- F/_ x ( A ( x e. C , y e. D |-> R ) B )
12	11	nfeq1 2322	. . . 4 |- F/ x ( A ( x e. C , y e. D |-> R ) B ) = S
13		nfmpo2 5921	. . . 4 |- F/_ y ( x e. C , y e. D |-> R )
14		nfcv 2312	. . . . . 6 |- F/_ y A
15		nfcv 2312	. . . . . 6 |- F/_ y B
16	14, 13, 15	nfov 5883	. . . . 5 |- F/_ y ( A ( x e. C , y e. D |-> R ) B )
17	16	nfeq1 2322	. . . 4 |- F/ y ( A ( x e. C , y e. D |-> R ) B ) = S
18	2, 3, 4, 7, 8, 12, 13, 17	ovmpodf 5984	. . 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 5859	. . 3 |- ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) )
21	20	eqeq1d 2179	. 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 156	1 |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = S ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141  (class class class)co 5853   e. cmpo 5855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858

This theorem is referenced by: (None)