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Theorem aecom-o 2232
Description: Commutation law for identical variable specifiers. The antecedent and consequent are true when  x and  y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 2055 using ax-c11 2220. Unlike axc11nfromc11 2258, this version does not require ax-5 1709. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)
Assertion
Ref Expression
aecom-o  |-  ( A. x  x  =  y  ->  A. y  y  =  x )

Proof of Theorem aecom-o
StepHypRef Expression
1 ax-c11 2220 . . 3  |-  ( A. x  x  =  y  ->  ( A. x  x  =  y  ->  A. y  x  =  y )
)
21pm2.43i 47 . 2  |-  ( A. x  x  =  y  ->  A. y  x  =  y )
3 equcomi 1798 . . 3  |-  ( x  =  y  ->  y  =  x )
43alimi 1638 . 2  |-  ( A. y  x  =  y  ->  A. y  y  =  x )
52, 4syl 16 1  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-c11 2220
This theorem depends on definitions:  df-bi 185  df-ex 1618
This theorem is referenced by:  aecoms-o  2233  naecoms-o  2259  aev-o  2263  ax12indalem  2277
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