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Theorem ax9o 1950
 Description: Show that the original axiom ax-9o 2138 can be derived from ax9 1949 and others. See ax9from9o 2148 for the rederivation of ax9 1949 from ax-9o 2138. Normally, ax9o 1950 should be used rather than ax-9o 2138, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax9o (x(x = yxφ) → φ)

Proof of Theorem ax9o
StepHypRef Expression
1 ax9 1949 . . 3 ¬ x ¬ x = y
2 con3 126 . . . 4 ((x = yxφ) → (¬ xφ → ¬ x = y))
32al2imi 1561 . . 3 (x(x = yxφ) → (x ¬ xφx ¬ x = y))
41, 3mtoi 169 . 2 (x(x = yxφ) → ¬ x ¬ xφ)
5 ax6o 1750 . 2 x ¬ xφφ)
64, 5syl 15 1 (x(x = yxφ) → φ)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540 This theorem is referenced by:  equsal  1960  spimt  1974  cbv1h  1978 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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