Theorem ovmpodv2 6052

Description: Alternate deduction version of ovmpo 6054, 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 2194	. . 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 2205	. . . . 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 5985	. . . 4 |- F/_ x ( x e. C , y e. D |-> R )
9		nfcv 2336	. . . . . 6 |- F/_ x A
10		nfcv 2336	. . . . . 6 |- F/_ x B
11	9, 8, 10	nfov 5948	. . . . 5 |- F/_ x ( A ( x e. C , y e. D |-> R ) B )
12	11	nfeq1 2346	. . . 4 |- F/ x ( A ( x e. C , y e. D |-> R ) B ) = S
13		nfmpo2 5986	. . . 4 |- F/_ y ( x e. C , y e. D |-> R )
14		nfcv 2336	. . . . . 6 |- F/_ y A
15		nfcv 2336	. . . . . 6 |- F/_ y B
16	14, 13, 15	nfov 5948	. . . . 5 |- F/_ y ( A ( x e. C , y e. D |-> R ) B )
17	16	nfeq1 2346	. . . 4 |- F/ y ( A ( x e. C , y e. D |-> R ) B ) = S
18	2, 3, 4, 7, 8, 12, 13, 17	ovmpodf 6050	. . 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 5924	. . 3 |- ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) )
21	20	eqeq1d 2202	. 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 1364   e. wcel 2164  (class class class)co 5918   e. cmpo 5920

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923

This theorem is referenced by: (None)