Theorem bj-axc10v 34975

Description: Version of axc10 2385 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axc10v	⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-axc10v

Step	Hyp	Ref	Expression
1		ax6v 1972	. . 3 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2		con3 153	. . . 4 ⊢ ((𝑥 = 𝑦 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝑥 = 𝑦))
3	2	al2imi 1818	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦))
4	1, 3	mtoi 198	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
5		axc7 2311	. 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 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by: (None)