Theorem hbe1a 2023

Description: Dual statement of hbe1 1495. (Contributed by Wolf Lammen, 15-Sep-2021.)

Assertion

Ref	Expression
hbe1a	⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)

Proof of Theorem hbe1a

Step	Hyp	Ref	Expression
1		nfa1 1541	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
2		nf3 1669	. . 3 ⊢ (Ⅎ𝑥∀𝑥𝜑 ↔ ∀𝑥(∃𝑥∀𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	mpbi 145	. 2 ⊢ ∀𝑥(∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
4	3	spi 1536	1 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1351  Ⅎwnf 1460  ∃wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  nf5-1  2024