Metamath Proof Explorer


Theorem bj-nnfe1

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

Ref Expression
Assertion bj-nnfe1 Ⅎ' 𝑥𝑥 𝜑

Proof

Step Hyp Ref Expression
1 bj-modal4e ( ∃ 𝑥𝑥 𝜑 → ∃ 𝑥 𝜑 )
2 hbe1 ( ∃ 𝑥 𝜑 → ∀ 𝑥𝑥 𝜑 )
3 df-bj-nnf ( Ⅎ' 𝑥𝑥 𝜑 ↔ ( ( ∃ 𝑥𝑥 𝜑 → ∃ 𝑥 𝜑 ) ∧ ( ∃ 𝑥 𝜑 → ∀ 𝑥𝑥 𝜑 ) ) )
4 1 2 3 mpbir2an Ⅎ' 𝑥𝑥 𝜑