Theorem axc10 2385

Description: Show that the original axiom ax-c10 36827 can be derived from ax6 2384 and axc7 2315 (on top of propositional calculus, ax-gen 1799, and ax-4 1813). See ax6fromc10 36837 for the rederivation of ax6 2384 from ax-c10 36827.

Normally, axc10 2385 should be used rather than ax-c10 36827, except by theorems specifically studying the latter's properties. See bj-axc10v 34902 for a weaker version requiring fewer axioms. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) Usage of this theorem is discouraged because it depends on ax-13 2372. (New usage is discouraged.)

Assertion

Ref	Expression
axc10	⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Proof of Theorem axc10

Step	Hyp	Ref	Expression
1		ax6 2384	. . 3 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2		con3 153	. . . 4 ⊢ ((𝑥 = 𝑦 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝑥 = 𝑦))
3	2	al2imi 1819	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦))
4	1, 3	mtoi 198	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
5		axc7 2315	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
6	4, 5	syl 17	1 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784

This theorem is referenced by:  spALT  41701