Theorem bj-axc10 36778

Description: Alternate proof of axc10 2390. Shorter. One can prove a version with DV (𝑥, 𝑦) without ax-13 2377, by using ax6ev 1969 instead of ax6e 2388. (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 2388	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
2		exim 1833	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥∀𝑥𝜑))
3	1, 2	mpi 20	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ∃𝑥∀𝑥𝜑)
4		axc7e 2319	. 2 ⊢ (∃𝑥∀𝑥𝜑 → 𝜑)
5	3, 4	syl 17	1 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2177  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779

This theorem is referenced by: (None)