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Theorem proplem2 13607
Description: Lemma for mndpropd 14414. (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 5881 . . 3  |-  ( x  =  X  ->  (
x F y )  =  ( X F y ) )
21eleq1d 2362 . 2  |-  ( x  =  X  ->  (
( x F y )  e.  C  <->  ( X F y )  e.  C ) )
3 oveq2 5882 . . 3  |-  ( y  =  Y  ->  ( X F y )  =  ( X F Y ) )
43eleq1d 2362 . 2  |-  ( y  =  Y  ->  (
( X F y )  e.  C  <->  ( X F Y )  e.  C
) )
52, 4rspc2va 2904 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 1632    e. wcel 1696   A.wral 2556  (class class class)co 5874
This theorem is referenced by:  mndpropd  14414  issubmnd  14417  submcl  14446  issubg2  14652  gass  14771  lmodprop2d  15703  lsspropd  15790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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