Theorem bj-nnfe1 37817

Description: See nfe1 2189. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nnfe1	⊢ Ⅎ'𝑥∃𝑥𝜑

Proof of Theorem bj-nnfe1

Step	Hyp	Ref	Expression
1		bj-modal4e 37749	. 2 ⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑)
2		hbe1 2181	. 2 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
3		df-bj-nnf 37759	. 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 1567  ∃wex 1811  Ⅎ'wnnf 37758

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-10 2179  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-bj-nnf 37759

This theorem is used by: (None)