 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  axc10 Structured version   Visualization version   GIF version

Theorem axc10 2414
 Description: Show that the original axiom ax-c10 34687 can be derived from ax6 2413 and axc7 2296 (on top of propositional calculus, ax-gen 1870, and ax-4 1885). See ax6fromc10 34697 for the rederivation of ax6 2413 from ax-c10 34687. Normally, axc10 2414 should be used rather than ax-c10 34687, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.)
Assertion
Ref Expression
axc10 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Proof of Theorem axc10
StepHypRef Expression
1 ax6 2413 . . 3 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2 con3 150 . . . 4 ((𝑥 = 𝑦 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝑥 = 𝑦))
32al2imi 1891 . . 3 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦))
41, 3mtoi 190 . 2 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
5 axc7 2296 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
64, 5syl 17 1 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
 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  ax-13 2408 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator