Theorem axc10 2420

Description: Show that the original axiom ax-c10 39701 can be derived from ax6 2419 and axc7 2353 (on top of propositional calculus, ax-gen 1828, and ax-4 1842). See ax6fromc10 39711 for the rederivation of ax6 2419 from ax-c10 39701.

Normally, axc10 2420 should be used rather than ax-c10 39701, except by theorems specifically studying the latter's properties. See bj-axc10v 37469 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 2407. (New usage is discouraged.)

Assertion

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

Proof of Theorem axc10

Step	Hyp	Ref	Expression
1		ax6 2419	. . 3 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2		con3 154	. . . 4 ⊢ ((𝑥 = 𝑦 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝑥 = 𝑦))
3	2	al2imi 1848	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦))
4	1, 3	mtoi 202	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
5		axc7 2353	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
6	4, 5	syl 18	1 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216  ax-13 2407

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  spALT  44968