Theorem bj-axc10 34965

Description: Alternate proof of axc10 2385. Shorter. One can prove a version with DV (𝑥, 𝑦) without ax-13 2372, by using ax6ev 1973 instead of ax6e 2383. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-axc10

Step	Hyp	Ref	Expression
1		ax6e 2383	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
2		exim 1836	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥∀𝑥𝜑))
3	1, 2	mpi 20	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ∃𝑥∀𝑥𝜑)
4		axc7e 2312	. 2 ⊢ (∃𝑥∀𝑥𝜑 → 𝜑)
5	3, 4	syl 17	1 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782

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  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783

This theorem is referenced by: (None)