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Theorem ax11b 1995
Description: A bidirectional version of ax11o 1994. (Contributed by NM, 30-Jun-2006.)
Assertion
Ref Expression
ax11b ((¬ x x = y x = y) → (φx(x = yφ)))

Proof of Theorem ax11b
StepHypRef Expression
1 ax11o 1994 . . 3 x x = y → (x = y → (φx(x = yφ))))
21imp 418 . 2 ((¬ x x = y x = y) → (φx(x = yφ)))
3 sp 1747 . . . 4 (x(x = yφ) → (x = yφ))
43com12 27 . . 3 (x = y → (x(x = yφ) → φ))
54adantl 452 . 2 ((¬ x x = y x = y) → (x(x = yφ) → φ))
62, 5impbid 183 1 ((¬ x x = y x = y) → (φx(x = yφ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 176   wa 358  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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545
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