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Theorem proplem2 13843
Description: Lemma for mndpropd 14650. (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 6029 . . 3  |-  ( x  =  X  ->  (
x F y )  =  ( X F y ) )
21eleq1d 2455 . 2  |-  ( x  =  X  ->  (
( x F y )  e.  C  <->  ( X F y )  e.  C ) )
3 oveq2 6030 . . 3  |-  ( y  =  Y  ->  ( X F y )  =  ( X F Y ) )
43eleq1d 2455 . 2  |-  ( y  =  Y  ->  (
( X F y )  e.  C  <->  ( X F Y )  e.  C
) )
52, 4rspc2va 3004 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 359    = wceq 1649    e. wcel 1717   A.wral 2651  (class class class)co 6022
This theorem is referenced by:  mndpropd  14650  issubmnd  14653  submcl  14682  issubg2  14888  gass  15007  lmodprop2d  15935  lsspropd  16022  off2  23899  ofcf  24284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-iota 5360  df-fv 5404  df-ov 6025
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