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Theorem aecom-o 2151
 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 1946 using ax-10o 2139. Unlike ax10from10o 2177, this version does not require ax-17 1616. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
Assertion
Ref Expression
aecom-o (x x = yy y = x)

Proof of Theorem aecom-o
StepHypRef Expression
1 ax-10o 2139 . . 3 (x x = y → (x x = yy x = y))
21pm2.43i 43 . 2 (x x = yy x = y)
3 equcomi 1679 . . 3 (x = yy = x)
43alimi 1559 . 2 (y x = yy y = x)
52, 4syl 15 1 (x x = yy y = x)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-10o 2139 This theorem depends on definitions:  df-bi 177  df-ex 1542 This theorem is referenced by:  aecoms-o  2152  naecoms-o  2178  aev-o  2182  ax11indalem  2197
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