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Theorem axc10 2405
 Description: Show that the original axiom ax-c10 36127 can be derived from ax6 2404 and axc7 2338 (on top of propositional calculus, ax-gen 1797, and ax-4 1811). See ax6fromc10 36137 for the rederivation of ax6 2404 from ax-c10 36127. Normally, axc10 2405 should be used rather than ax-c10 36127, except by theorems specifically studying the latter's properties. Usage of this theorem has been discouraged later on to avoid ax-13 2392 propagation. Check out bj-axc10v 34174 for a weaker version requiring less axioms. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc10 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Proof of Theorem axc10
StepHypRef Expression
1 ax6 2404 . . 3 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2 con3 156 . . . 4 ((𝑥 = 𝑦 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝑥 = 𝑦))
32al2imi 1817 . . 3 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦))
41, 3mtoi 202 . 2 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
5 axc7 2338 . 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 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-12 2179  ax-13 2392 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by:  spALT  40826
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