Theorem bj-nnfa1 37468

Description: See nfa1 2189. (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 2182	. 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
2		bj-modal4 37400	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
3		df-bj-nnf 37411	. 2 ⊢ (Ⅎ'𝑥∀𝑥𝜑 ↔ ((∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) ∧ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)))
4	1, 2, 3	mpbir2an 724	1 ⊢ Ⅎ'𝑥∀𝑥𝜑

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812  Ⅎ'wnnf 37410

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-bj-nnf 37411

This theorem is used by: (None)