Theorem ovmpodv2 6004

Description: Alternate deduction version of ovmpo 6006, 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 2178	. . 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 2189	. . . . 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 5938	. . . 4 |- F/_ x ( x e. C , y e. D |-> R )
9		nfcv 2319	. . . . . 6 |- F/_ x A
10		nfcv 2319	. . . . . 6 |- F/_ x B
11	9, 8, 10	nfov 5901	. . . . 5 |- F/_ x ( A ( x e. C , y e. D |-> R ) B )
12	11	nfeq1 2329	. . . 4 |- F/ x ( A ( x e. C , y e. D |-> R ) B ) = S
13		nfmpo2 5939	. . . 4 |- F/_ y ( x e. C , y e. D |-> R )
14		nfcv 2319	. . . . . 6 |- F/_ y A
15		nfcv 2319	. . . . . 6 |- F/_ y B
16	14, 13, 15	nfov 5901	. . . . 5 |- F/_ y ( A ( x e. C , y e. D |-> R ) B )
17	16	nfeq1 2329	. . . 4 |- F/ y ( A ( x e. C , y e. D |-> R ) B ) = S
18	2, 3, 4, 7, 8, 12, 13, 17	ovmpodf 6002	. . 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 5877	. . 3 |- ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) )
21	20	eqeq1d 2186	. 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 1353   e. wcel 2148  (class class class)co 5871   e. cmpo 5873

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-setind 4535

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-iota 5176  df-fun 5216  df-fv 5222  df-ov 5874  df-oprab 5875  df-mpo 5876

This theorem is referenced by: (None)