Theorem ax4fromc4 40044

Description: Rederivation of Axiom ax-4 1841 from ax-c4 40034, ax-c5 40033, ax-gen 1827 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 2351 for the derivation of ax-c4 40034 from ax-4 1841. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-4 1841 instead. (New usage is discouraged.)

Assertion

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

Proof of Theorem ax4fromc4

Step	Hyp	Ref	Expression
1		ax-c4 40034	. . 3 ⊢ (∀𝑥(∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) → (∀𝑥(𝜑 → 𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓)))
2		ax-c5 40033	. . . 4 ⊢ (∀𝑥𝜑 → 𝜑)
3		ax-c5 40033	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓))
4	2, 3	syl5 35	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
5	1, 4	mpg 1829	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓))
6		ax-c4 40034	. 2 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
7	5, 6	syl 18	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1827  ax-c5 40033  ax-c4 40034

This theorem is used by: (None)