Theorem hbe1a 2039

Description: Dual statement of hbe1 1506. (Contributed by Wolf Lammen, 15-Sep-2021.)

Assertion

Ref	Expression
hbe1a	⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)

Proof of Theorem hbe1a

Step	Hyp	Ref	Expression
1		nfa1 1552	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
2		nf3 1680	. . 3 ⊢ (Ⅎ𝑥∀𝑥𝜑 ↔ ∀𝑥(∃𝑥∀𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	mpbi 145	. 2 ⊢ ∀𝑥(∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
4	3	spi 1547	1 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1362  Ⅎwnf 1471  ∃wex 1503

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  nf5-1  2040