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Theorem ax4fromc4 33626
Description: Rederivation of axiom ax-4 1734 from ax-c4 33616, ax-c5 33615, ax-gen 1719 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 2131 for the derivation of ax-c4 33616 from ax-4 1734. (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 33616 . . 3 (∀𝑥(∀𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓)) → (∀𝑥(𝜑𝜓) → ∀𝑥(∀𝑥𝜑𝜓)))
2 ax-c5 33615 . . . 4 (∀𝑥𝜑𝜑)
3 ax-c5 33615 . . . 4 (∀𝑥(𝜑𝜓) → (𝜑𝜓))
42, 3syl5 34 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))
51, 4mpg 1721 . 2 (∀𝑥(𝜑𝜓) → ∀𝑥(∀𝑥𝜑𝜓))
6 ax-c4 33616 . 2 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
75, 6syl 17 1 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1719  ax-c5 33615  ax-c4 33616
This theorem is referenced by: (None)
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