Theorem bj-nnfe1 37023

Description: See nfe1 2156. (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 36960	. 2 ⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑)
2		hbe1 2149	. 2 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
3		df-bj-nnf 36970	. 2 ⊢ (Ⅎ'𝑥∃𝑥𝜑 ↔ ((∃𝑥∃𝑥𝜑 → ∃𝑥𝜑) ∧ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)))
4	1, 2, 3	mpbir2an 712	1 ⊢ Ⅎ'𝑥∃𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1781  Ⅎ'wnnf 36969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-10 2147  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-bj-nnf 36970

This theorem is referenced by: (None)