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Theorem issubgoilem 20992
Description: Lemma for issubgoi 20993. (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 5881 . . 3  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
2 oveq1 5881 . . 3  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
31, 2eqeq12d 2310 . 2  |-  ( x  =  A  ->  (
( x H y )  =  ( x G y )  <->  ( A H y )  =  ( A G y ) ) )
4 oveq2 5882 . . 3  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
5 oveq2 5882 . . 3  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
64, 5eqeq12d 2310 . 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 2864 1  |-  ( ( A  e.  Y  /\  B  e.  Y )  ->  ( A H B )  =  ( A G B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696  (class class class)co 5874
This theorem is referenced by:  issubgoi  20993
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-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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