Theorem ax4fromc4 33197

Description: Rederivation of axiom ax-4 1728 from ax-c4 33187, ax-c5 33186, ax-gen 1713 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 2115 for the derivation of ax-c4 33187 from ax-4 1728. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax4fromc4	⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem ax4fromc4

Step	Hyp	Ref	Expression
1		ax-c4 33187	. . 3 ⊢ (∀𝑥(∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) → (∀𝑥(𝜑 → 𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓)))
2		ax-c5 33186	. . . 4 ⊢ (∀𝑥𝜑 → 𝜑)
3		ax-c5 33186	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓))
4	2, 3	syl5 33	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
5	1, 4	mpg 1715	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓))
6		ax-c4 33187	. 2 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
7	5, 6	syl 17	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1713  ax-c5 33186  ax-c4 33187

This theorem is referenced by: (None)