Theorem hbe1a 2181

Description: Dual statement of hbe1 2180. Modified version of axc7e 2349 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 1813	. 2 ⊢ (∃𝑥∀𝑥𝜑 ↔ ¬ ∀𝑥 ¬ ∀𝑥𝜑)
2		hbn1 2179	. . 3 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3	2	con1i 148	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
4	1, 3	sylbi 220	1 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-10 2178

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

This theorem is used by:  nf5-1  2182  nfexa2  2212  axc7e  2349  nfeqf2  2407  bj-19.41al  37528  bj-subst  37530  bj-modal4  37588  bj-wnf2  37592  bj-substax12  37596  bj-nnfa1  37656