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Theorem axc4 1991
Description: Show that the original axiom ax-c4 33062 can be derived from ax-4 1713 (alim 1714), ax-10 1966 (hbn1 1967), sp 1990 and propositional calculus. See ax4fromc4 33072 for the rederivation of ax-4 1713 from ax-c4 33062.

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
axc4 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem axc4
StepHypRef Expression
1 sp 1990 . . . 4 (∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝜑)
21con2i 132 . . 3 (∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
3 hbn1 1967 . . 3 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑)
4 hbn1 1967 . . . . 5 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
54con1i 142 . . . 4 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
65alimi 1715 . . 3 (∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝑥𝜑)
72, 3, 63syl 18 . 2 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
8 alim 1714 . 2 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝑥𝜑 → ∀𝑥𝜓))
97, 8syl5 33 1 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-12 1983
This theorem depends on definitions:  df-bi 195  df-ex 1695
This theorem is referenced by:  axc5c4c711  37506
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