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Theorem axc10 2288
 Description: Show that the original axiom ax-c10 34490 can be derived from ax6 2287 and axc7 2170 (on top of propositional calculus, ax-gen 1762, and ax-4 1777). See ax6fromc10 34500 for the rederivation of ax6 2287 from ax-c10 34490. Normally, axc10 2288 should be used rather than ax-c10 34490, 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 2287 . . 3 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2 con3 149 . . . 4 ((𝑥 = 𝑦 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝑥 = 𝑦))
32al2imi 1783 . . 3 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦))
41, 3mtoi 190 . 2 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
5 axc7 2170 . 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 1521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745 This theorem is referenced by: (None)
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