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Theorem hbae-x12 29109
Description: Experiment to study ax12o 1875. Weak version of hbae 1893. Does not use sp 1716, ax9 1889, ax10 1884, or ax12o 1875 but allows ax9v 1636. (Contributed by NM, 7-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
hbae-x12  |-  ( A. x  x  =  y  ->  A. y A. x  x  =  y )
Distinct variable group:    x, y

Proof of Theorem hbae-x12
StepHypRef Expression
1 hba1 1719 . . 3  |-  ( A. x  x  =  y  ->  A. x A. x  x  =  y )
2 ax10lem3 1878 . . 3  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
31, 2alrimih 1552 . 2  |-  ( A. x  x  =  y  ->  A. x A. y 
y  =  x )
4 equcomi 1646 . . 3  |-  ( y  =  x  ->  x  =  y )
542alimi 1547 . 2  |-  ( A. x A. y  y  =  x  ->  A. x A. y  x  =  y )
6 ax-7 1708 . 2  |-  ( A. x A. y  x  =  y  ->  A. y A. x  x  =  y )
73, 5, 63syl 18 1  |-  ( A. x  x  =  y  ->  A. y A. x  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527
This theorem is referenced by:  hbnae-x12  29110
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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