Theorem axc4 2130

Description: Show that the original axiom ax-c4 34169 can be derived from ax-4 1737 (alim 1738), ax-10 2019 (hbn1 2020), sp 2053 and propositional calculus. See ax4fromc4 34179 for the rederivation of ax-4 1737 from ax-c4 34169.

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 2053	. . . 4 ⊢ (∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝜑)
2	1	con2i 134	. . 3 ⊢ (∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
3		hbn1 2020	. . 3 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑)
4		hbn1 2020	. . . . 5 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
5	4	con1i 144	. . . 4 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
6	5	alimi 1739	. . 3 ⊢ (∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
7	2, 3, 6	3syl 18	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
8		alim 1738	. 2 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥∀𝑥𝜑 → ∀𝑥𝜓))
9	7, 8	syl5 34	1 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  axc5c4c711  38602