Theorem hbe1a 2003

Description: Dual statement of hbe1 1475. (Contributed by Wolf Lammen, 15-Sep-2021.)

Assertion

Ref	Expression
hbe1a	⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)

Proof of Theorem hbe1a

Step	Hyp	Ref	Expression
1		nfa1 1521	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
2		nf3 1649	. . 3 ⊢ (Ⅎ𝑥∀𝑥𝜑 ↔ ∀𝑥(∃𝑥∀𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	mpbi 144	. 2 ⊢ ∀𝑥(∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
4	3	spi 1516	1 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1333  Ⅎwnf 1440  ∃wex 1472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1441

This theorem is referenced by:  nf5-1  2004