Theorem axc7e 2337

Description: Abbreviated version of axc7 2336 using the existential quantifier. Corresponds to the dual of Axiom (B) of modal logic. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Jul-2022.)

Assertion

Ref	Expression
axc7e	⊢ (∃𝑥∀𝑥𝜑 → 𝜑)

Proof of Theorem axc7e

Step	Hyp	Ref	Expression
1		hbe1a 2148	. 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
2	1	19.21bi 2188	1 ⊢ (∃𝑥∀𝑥𝜑 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-ex 1781

This theorem is referenced by:  bj-19.12  34092  bj-axc10  34107