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Theorem proplem 13917
Description: Lemma for mndpropd 14723. (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 2800 . 2  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) )
3 oveq1 6090 . . . 4  |-  ( x  =  X  ->  (
x F y )  =  ( X F y ) )
4 oveq1 6090 . . . 4  |-  ( x  =  X  ->  (
x G y )  =  ( X G y ) )
53, 4eqeq12d 2452 . . 3  |-  ( x  =  X  ->  (
( x F y )  =  ( x G y )  <->  ( X F y )  =  ( X G y ) ) )
6 oveq2 6091 . . . 4  |-  ( y  =  Y  ->  ( X F y )  =  ( X F Y ) )
7 oveq2 6091 . . . 4  |-  ( y  =  Y  ->  ( X G y )  =  ( X G Y ) )
86, 7eqeq12d 2452 . . 3  |-  ( y  =  Y  ->  (
( X F y )  =  ( X G y )  <->  ( X F Y )  =  ( X G Y ) ) )
95, 8rspc2v 3060 . 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 457 1  |-  ( (
ph  /\  ( X  e.  A  /\  Y  e.  B ) )  -> 
( X F Y )  =  ( X G Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707  (class class class)co 6083
This theorem is referenced by:  mndpropd  14723  grpidpropd  14724  grpsubpropd2  14892  cmnpropd  15423  rngpropd  15697  lmodprop2d  16008  lsspropd  16095  lmhmpropd  16147  lbspropd  16173  assapropd  16388  asclpropd  16406  psrplusgpropd  16631  phlpropd  16888  gsumpropd2lem  24222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086
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