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Theorem axc10 2143
Description: Show that the original axiom ax-c10 32691 can be derived from ax6 2142 and others. See ax6fromc10 32701 for the rederivation of ax6 2142 from ax-c10 32691.

Normally, axc10 2143 should be used rather than ax-c10 32691, 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 2142 . . 3 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2 con3 143 . . . 4 ((𝑥 = 𝑦 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝑥 = 𝑦))
32al2imi 1716 . . 3 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦))
41, 3mtoi 185 . 2 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
5 axc7 1992 . 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 1466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-12 1983  ax-13 2137
This theorem depends on definitions:  df-bi 192  df-an 380  df-ex 1693
This theorem is referenced by: (None)
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