Theorem axc7e 2317

Description: Abbreviated version of axc7 2316 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 2144	. 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
2	1	19.21bi 2186	1 ⊢ (∃𝑥∀𝑥𝜑 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1541  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-ex 1788

This theorem is referenced by:  bj-19.12  34680  bj-axc10  34702