Theorem hbe1a 2140

Description: Dual statement of hbe1 2139. Modified version of axc7e 2311 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 1782	. 2 ⊢ (∃𝑥∀𝑥𝜑 ↔ ¬ ∀𝑥 ¬ ∀𝑥𝜑)
2		hbn1 2138	. . 3 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	2	con1i 147	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
4	1, 3	sylbi 216	1 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-10 2137

This theorem depends on definitions:  df-bi 206  df-ex 1782

This theorem is referenced by:  nf5-1  2141  axc7e  2311  nfeqf2  2376  bj-19.41al  35531  bj-subst  35533  bj-modal4  35587  bj-wnf2  35591  bj-substax12  35594  bj-nnfa1  35632