Theorem axc7e 2354

Description: Abbreviated version of axc7 2353 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 2182	. 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
2	1	19.21bi 2228	1 ⊢ (∃𝑥∀𝑥𝜑 → 𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  bj-19.12  37389  bj-axc10  37459