Theorem axc4 2357

Description: Show that the original axiom ax-c4 39698 can be derived from ax-4 1842 (alim 1843), ax-10 2179 (hbn1 2180), sp 2222 and propositional calculus. See ax4fromc4 39708 for the rederivation of ax-4 1842 from ax-c4 39698.

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(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem axc4

Step	Hyp	Ref	Expression
1		sp 2222	. . . 4 ⊢ (∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝜑)
2	1	con2i 140	. . 3 ⊢ (∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
3		hbn1 2180	. . 3 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑)
4		hbn1 2180	. . . . 5 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
5	4	con1i 148	. . . 4 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
6	5	alimi 1844	. . 3 ⊢ (∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
7	2, 3, 6	3syl 19	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
8		alim 1843	. 2 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥∀𝑥𝜑 → ∀𝑥𝜓))
9	7, 8	syl5 35	1 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  axc5c4c711  45151