Theorem axc10 2399

Description: Show that the original axiom ax-c10 36016 can be derived from ax6 2398 and axc7 2332 (on top of propositional calculus, ax-gen 1792, and ax-4 1806). See ax6fromc10 36026 for the rederivation of ax6 2398 from ax-c10 36016.

Normally, axc10 2399 should be used rather than ax-c10 36016, except by theorems specifically studying the latter's properties. Usage of this theorem has been discouraged later on to avoid ax-13 2386 propagation. Check out bj-axc10v 34110 for a weaker version requiring less axioms. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem axc10

Step	Hyp	Ref	Expression
1		ax6 2398	. . 3 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2		con3 156	. . . 4 ⊢ ((𝑥 = 𝑦 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝑥 = 𝑦))
3	2	al2imi 1812	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦))
4	1, 3	mtoi 201	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
5		axc7 2332	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
6	4, 5	syl 17	1 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-12 2173  ax-13 2386

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777

This theorem is referenced by:  spALT  40547