Theorem axc7e 2258

Description: Abbreviated version of axc7 2257 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 2082	. 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
2	1	19.21bi 2117	1 ⊢ (∃𝑥∀𝑥𝜑 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1505  ∃wex 1742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-12 2106

This theorem depends on definitions:  df-bi 199  df-ex 1743

This theorem is referenced by:  19.9ht  2260  bj-axc10  33560  bj-19.12  33786