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

Theorem axc4 2115
 Description: Show that the original axiom ax-c4 33187 can be derived from ax-4 1728 (alim 1729), ax-10 2006 (hbn1 2007), sp 2041 and propositional calculus. See ax4fromc4 33197 for the rederivation of ax-4 1728 from ax-c4 33187. 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 2041 . . . 4 (∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝜑)
21con2i 133 . . 3 (∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
3 hbn1 2007 . . 3 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑)
4 hbn1 2007 . . . . 5 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
54con1i 143 . . . 4 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
65alimi 1730 . . 3 (∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝑥𝜑)
72, 3, 63syl 18 . 2 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
8 alim 1729 . 2 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝑥𝜑 → ∀𝑥𝜓))
97, 8syl5 33 1 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696 This theorem is referenced by:  axc5c4c711  37624
 Copyright terms: Public domain W3C validator