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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(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem ax4fromc4
StepHypRef Expression
1 ax-c4 33187 . . 3 (∀𝑥(∀𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓)) → (∀𝑥(𝜑𝜓) → ∀𝑥(∀𝑥𝜑𝜓)))
2 ax-c5 33186 . . . 4 (∀𝑥𝜑𝜑)
3 ax-c5 33186 . . . 4 (∀𝑥(𝜑𝜓) → (𝜑𝜓))
42, 3syl5 33 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))
51, 4mpg 1715 . 2 (∀𝑥(𝜑𝜓) → ∀𝑥(∀𝑥𝜑𝜓))
6 ax-c4 33187 . 2 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
75, 6syl 17 1 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
 Colors of variables: wff setvar class 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)
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