Theorem ovmpt2dv 5633

Description: Alternate deduction version of ovmpt2 5636, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Hypotheses

Ref	Expression
ovmpt2df.1	|- ( ph -> A e. C )
ovmpt2df.2	|- ( ( ph /\ x = A ) -> B e. D )
ovmpt2df.3	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
ovmpt2df.4	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = R -> ps ) )

Assertion

Ref	Expression
ovmpt2dv	|- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )

Distinct variable groups:   x, y, A   y, B   ph, x, y   x, F, y   ps, x, y

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

Proof of Theorem ovmpt2dv

Step	Hyp	Ref	Expression
1		ovmpt2df.1	. 2 |- ( ph -> A e. C )
2		ovmpt2df.2	. 2 |- ( ( ph /\ x = A ) -> B e. D )
3		ovmpt2df.3	. 2 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
4		ovmpt2df.4	. 2 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = R -> ps ) )
5		nfcv 2178	. 2 |- F/_ x F
6		nfv 1421	. 2 |- F/ x ps
7		nfcv 2178	. 2 |- F/_ y F
8		nfv 1421	. 2 |- F/ y ps
9	1, 2, 3, 4, 5, 6, 7, 8	ovmpt2df 5632	1 |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393  (class class class)co 5512   |-> cmpt2 5514

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517

This theorem is referenced by: (None)