Theorem bj-nnfa1 34107

Description: See nfa1 2154. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nnfa1	⊢ Ⅎ'𝑥∀𝑥𝜑

Proof of Theorem bj-nnfa1

Step	Hyp	Ref	Expression
1		hbe1a 2147	. 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
2		bj-modal4 34067	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
3		df-bj-nnf 34075	. 2 ⊢ (Ⅎ'𝑥∀𝑥𝜑 ↔ ((∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) ∧ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)))
4	1, 2, 3	mpbir2an 709	1 ⊢ Ⅎ'𝑥∀𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1534  ∃wex 1779  Ⅎ'wnnf 34074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-bj-nnf 34075

This theorem is referenced by: (None)