MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  proplem Unicode version

Theorem proplem 13608
Description: Lemma for mndpropd 14414. (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 2648 . 2  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) )
3 oveq1 5881 . . . 4  |-  ( x  =  X  ->  (
x F y )  =  ( X F y ) )
4 oveq1 5881 . . . 4  |-  ( x  =  X  ->  (
x G y )  =  ( X G y ) )
53, 4eqeq12d 2310 . . 3  |-  ( x  =  X  ->  (
( x F y )  =  ( x G y )  <->  ( X F y )  =  ( X G y ) ) )
6 oveq2 5882 . . . 4  |-  ( y  =  Y  ->  ( X F y )  =  ( X F Y ) )
7 oveq2 5882 . . . 4  |-  ( y  =  Y  ->  ( X G y )  =  ( X G Y ) )
86, 7eqeq12d 2310 . . 3  |-  ( y  =  Y  ->  (
( X F y )  =  ( X G y )  <->  ( X F Y )  =  ( X G Y ) ) )
95, 8rspc2v 2903 . 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 455 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 358    = wceq 1632    e. wcel 1696   A.wral 2556  (class class class)co 5874
This theorem is referenced by:  mndpropd  14414  grpidpropd  14415  grpsubpropd2  14583  cmnpropd  15114  rngpropd  15388  lmodprop2d  15703  lsspropd  15790  lmhmpropd  15842  lbspropd  15868  assapropd  16083  asclpropd  16101  psrplusgpropd  16329  phlpropd  16575  gsumpropd2lem  23394
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
  Copyright terms: Public domain W3C validator