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Theorem ax4fromc4 34695
Description: Rederivation of axiom ax-4 1885 from ax-c4 34685, ax-c5 34684, ax-gen 1870 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 2294 for the derivation of ax-c4 34685 from ax-4 1885. (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 34685 . . 3 (∀𝑥(∀𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓)) → (∀𝑥(𝜑𝜓) → ∀𝑥(∀𝑥𝜑𝜓)))
2 ax-c5 34684 . . . 4 (∀𝑥𝜑𝜑)
3 ax-c5 34684 . . . 4 (∀𝑥(𝜑𝜓) → (𝜑𝜓))
42, 3syl5 34 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))
51, 4mpg 1872 . 2 (∀𝑥(𝜑𝜓) → ∀𝑥(∀𝑥𝜑𝜓))
6 ax-c4 34685 . 2 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
75, 6syl 17 1 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1870  ax-c5 34684  ax-c4 34685
This theorem is referenced by: (None)
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