Theorem hbe1a 2145

Description: Dual statement of hbe1 2144. Modified version of axc7e 2319 with a universally quantified consequent. (Contributed by Wolf Lammen, 15-Sep-2021.)

Assertion

Ref	Expression
hbe1a	⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)

Proof of Theorem hbe1a

Step	Hyp	Ref	Expression
1		df-ex 1780	. 2 ⊢ (∃𝑥∀𝑥𝜑 ↔ ¬ ∀𝑥 ¬ ∀𝑥𝜑)
2		hbn1 2143	. . 3 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	2	con1i 147	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
4	1, 3	sylbi 217	1 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-10 2142

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by:  nf5-1  2146  axc7e  2319  nfeqf2  2382  bj-19.41al  36682  bj-subst  36684  bj-modal4  36737  bj-wnf2  36741  bj-substax12  36744  bj-nnfa1  36782