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Theorem ax5o 1749
 Description: Show that the original axiom ax-5o 2136 can be derived from ax-5 1557 and others. See ax5 2146 for the rederivation of ax-5 1557 from ax-5o 2136. Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax5o (x(xφψ) → (xφxψ))

Proof of Theorem ax5o
StepHypRef Expression
1 sp 1747 . . . 4 (x ¬ xφ → ¬ xφ)
21con2i 112 . . 3 (xφ → ¬ x ¬ xφ)
3 hbn1 1730 . . 3 x ¬ xφx ¬ x ¬ xφ)
4 hbn1 1730 . . . . 5 xφx ¬ xφ)
54con1i 121 . . . 4 x ¬ xφxφ)
65alimi 1559 . . 3 (x ¬ x ¬ xφxxφ)
72, 3, 63syl 18 . 2 (xφxxφ)
8 ax-5 1557 . 2 (x(xφψ) → (xxφxψ))
97, 8syl5 28 1 (x(xφψ) → (xφxψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → 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-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542 This theorem is referenced by: (None)
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