Theorem bj-axc10 37450

Description: Alternate proof of axc10 2420. Shorter. One can prove a version with DV (𝑥, 𝑦) without ax-13 2407, by using ax6ev 2002 instead of ax6e 2418. (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 2418	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
2		exim 1867	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥∀𝑥𝜑))
3	1, 2	mpi 21	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ∃𝑥∀𝑥𝜑)
4		axc7e 2354	. 2 ⊢ (∃𝑥∀𝑥𝜑 → 𝜑)
5	3, 4	syl 18	1 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812

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: (None)