Theorem bj-axc10 31894

Description: Alternate (shorter) proof of axc10 2240. (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 2238	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
2		exim 1751	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥∀𝑥𝜑))
3	1, 2	mpi 20	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ∃𝑥∀𝑥𝜑)
4		axc7e 2118	. 2 ⊢ (∃𝑥∀𝑥𝜑 → 𝜑)
5	3, 4	syl 17	1 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by: (None)