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Theorem proplem 14649
Description: Lemma for mndpropd 15467. (Contributed by Mario Carneiro, 6-Dec-2014.)
Hypothesis
Ref Expression
proplem.1  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x F y )  =  ( x G y ) )
Assertion
Ref Expression
proplem  |-  ( (
ph  /\  ( X  e.  A  /\  Y  e.  B ) )  -> 
( X F Y )  =  ( X G Y ) )
Distinct variable groups:    x, y, A    x, B, y    x, F, y    ph, x, y   
y, Y    x, G, y    x, X, y
Allowed substitution hint:    Y( x)

Proof of Theorem proplem
StepHypRef Expression
1 proplem.1 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x F y )  =  ( x G y ) )
21ralrimivva 2829 . 2  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) )
3 oveq1 6119 . . . 4  |-  ( x  =  X  ->  (
x F y )  =  ( X F y ) )
4 oveq1 6119 . . . 4  |-  ( x  =  X  ->  (
x G y )  =  ( X G y ) )
53, 4eqeq12d 2457 . . 3  |-  ( x  =  X  ->  (
( x F y )  =  ( x G y )  <->  ( X F y )  =  ( X G y ) ) )
6 oveq2 6120 . . . 4  |-  ( y  =  Y  ->  ( X F y )  =  ( X F Y ) )
7 oveq2 6120 . . . 4  |-  ( y  =  Y  ->  ( X G y )  =  ( X G Y ) )
86, 7eqeq12d 2457 . . 3  |-  ( y  =  Y  ->  (
( X F y )  =  ( X G y )  <->  ( X F Y )  =  ( X G Y ) ) )
95, 8rspc2v 3100 . 2  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y )  ->  ( X F Y )  =  ( X G Y ) ) )
102, 9mpan9 469 1  |-  ( (
ph  /\  ( X  e.  A  /\  Y  e.  B ) )  -> 
( X F Y )  =  ( X G Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736  (class class class)co 6112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-iota 5402  df-fv 5447  df-ov 6115
This theorem is referenced by:  mndpropd  15467  grpidpropd  15468  gsumpropd2lem  15526  grpsubpropd2  15648  cmnpropd  16307  rngpropd  16698  lmodprop2d  17029  lsspropd  17120  lmhmpropd  17176  lbspropd  17202  assapropd  17420  asclpropd  17438  psrplusgpropd  17712  phlpropd  18106
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