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Theorem ovmpt2dv 5980
Description: Alternate deduction version of ovmpt2 5983, 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
StepHypRef 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 2419 . 2  |-  F/_ x F
6 nfv 1605 . 2  |-  F/ x ps
7 nfcv 2419 . 2  |-  F/_ y F
8 nfv 1605 . 2  |-  F/ y ps
91, 2, 3, 4, 5, 6, 7, 8ovmpt2df 5979 1  |-  ( ph  ->  ( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684  (class class class)co 5858    e. cmpt2 5860
This theorem is referenced by:  xpcco  13957  curf12  14001  curf2  14003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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