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Theorem issubgoilem 21899
Description: Lemma for issubgoi 21900. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
issubgoilem.1  |-  ( ( x  e.  Y  /\  y  e.  Y )  ->  ( x H y )  =  ( x G y ) )
Assertion
Ref Expression
issubgoilem  |-  ( ( A  e.  Y  /\  B  e.  Y )  ->  ( A H B )  =  ( A G B ) )
Distinct variable groups:    x, A, y    y, B    x, G, y    x, H, y    x, Y, y
Allowed substitution hint:    B( x)

Proof of Theorem issubgoilem
StepHypRef Expression
1 oveq1 6090 . . 3  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
2 oveq1 6090 . . 3  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
31, 2eqeq12d 2452 . 2  |-  ( x  =  A  ->  (
( x H y )  =  ( x G y )  <->  ( A H y )  =  ( A G y ) ) )
4 oveq2 6091 . . 3  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
5 oveq2 6091 . . 3  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
64, 5eqeq12d 2452 . 2  |-  ( y  =  B  ->  (
( A H y )  =  ( A G y )  <->  ( A H B )  =  ( A G B ) ) )
7 issubgoilem.1 . 2  |-  ( ( x  e.  Y  /\  y  e.  Y )  ->  ( x H y )  =  ( x G y ) )
83, 6, 7vtocl2ga 3021 1  |-  ( ( A  e.  Y  /\  B  e.  Y )  ->  ( A H B )  =  ( A G B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726  (class class class)co 6083
This theorem is referenced by:  issubgoi  21900
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-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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