Theorem axc10 2419

Description: Show that the original axiom ax-c10 39517 can be derived from ax6 2418 and axc7 2352 (on top of propositional calculus, ax-gen 1818, and ax-4 1832). See ax6fromc10 39527 for the rederivation of ax6 2418 from ax-c10 39517.

Normally, axc10 2419 should be used rather than ax-c10 39517, except by theorems specifically studying the latter's properties. See bj-axc10v 37285 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 2406. (New usage is discouraged.)

Assertion

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

Proof of Theorem axc10

Step	Hyp	Ref	Expression
1		ax6 2418	. . 3 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2		con3 154	. . . 4 ⊢ ((𝑥 = 𝑦 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝑥 = 𝑦))
3	2	al2imi 1838	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦))
4	1, 3	mtoi 202	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
5		axc7 2352	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
6	4, 5	syl 18	1 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-10 2178  ax-12 2215  ax-13 2406

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by:  spALT  44784