MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axc4 Structured version   Visualization version   GIF version

Theorem axc4 2294
Description: Show that the original axiom ax-c4 34691 can be derived from ax-4 1885 (alim 1886), ax-10 2174 (hbn1 2175), sp 2207 and propositional calculus. See ax4fromc4 34701 for the rederivation of ax-4 1885 from ax-c4 34691.

Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.)

Assertion
Ref Expression
axc4 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem axc4
StepHypRef Expression
1 sp 2207 . . . 4 (∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝜑)
21con2i 136 . . 3 (∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
3 hbn1 2175 . . 3 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑)
4 hbn1 2175 . . . . 5 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
54con1i 146 . . . 4 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
65alimi 1887 . . 3 (∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝑥𝜑)
72, 3, 63syl 18 . 2 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
8 alim 1886 . 2 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝑥𝜑 → ∀𝑥𝜓))
97, 8syl5 34 1 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-ex 1853
This theorem is referenced by:  axc5c4c711  39128
  Copyright terms: Public domain W3C validator