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Theorem proplem2 13591
Description: Lemma for mndpropd 14398. (Contributed by Mario Carneiro, 6-Dec-2014.)
Assertion
Ref Expression
proplem2  |-  ( ( ( X  e.  A  /\  Y  e.  B
)  /\  A. x  e.  A  A. y  e.  B  ( x F y )  e.  C )  ->  ( X F Y )  e.  C )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, F, y   
y, Y    x, X, y
Allowed substitution hint:    Y( x)

Proof of Theorem proplem2
StepHypRef Expression
1 oveq1 5865 . . 3  |-  ( x  =  X  ->  (
x F y )  =  ( X F y ) )
21eleq1d 2349 . 2  |-  ( x  =  X  ->  (
( x F y )  e.  C  <->  ( X F y )  e.  C ) )
3 oveq2 5866 . . 3  |-  ( y  =  Y  ->  ( X F y )  =  ( X F Y ) )
43eleq1d 2349 . 2  |-  ( y  =  Y  ->  (
( X F y )  e.  C  <->  ( X F Y )  e.  C
) )
52, 4rspc2va 2891 1  |-  ( ( ( X  e.  A  /\  Y  e.  B
)  /\  A. x  e.  A  A. y  e.  B  ( x F y )  e.  C )  ->  ( X F Y )  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543  (class class class)co 5858
This theorem is referenced by:  mndpropd  14398  issubmnd  14401  submcl  14430  issubg2  14636  gass  14755  lmodprop2d  15687  lsspropd  15774
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  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-iota 5219  df-fv 5263  df-ov 5861
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