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Theorem axc4 2126
Description: Show that the original axiom ax-c4 33646 can be derived from ax-4 1734 (alim 1735), ax-10 2016 (hbn1 2017), sp 2051 and propositional calculus. See ax4fromc4 33656 for the rederivation of ax-4 1734 from ax-c4 33646.

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 2051 . . . 4 (∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝜑)
21con2i 134 . . 3 (∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
3 hbn1 2017 . . 3 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑)
4 hbn1 2017 . . . . 5 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
54con1i 144 . . . 4 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
65alimi 1736 . . 3 (∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝑥𝜑)
72, 3, 63syl 18 . 2 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
8 alim 1735 . 2 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝑥𝜑 → ∀𝑥𝜓))
97, 8syl5 34 1 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-ex 1702
This theorem is referenced by:  axc5c4c711  38081
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