Theorem axc7e 2118

Description: Abbreviated version of axc7 2117 using the existential quantifier. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
axc7e	⊢ (∃𝑥∀𝑥𝜑 → 𝜑)

Proof of Theorem axc7e

Step	Hyp	Ref	Expression
1		hbe1a 2009	. 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
2		sp 2041	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	syl 17	1 ⊢ (∃𝑥∀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

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

This theorem is referenced by:  19.9ht  2128  bj-axc10  31894