Metamath Proof Explorer


Theorem bj-nfnnfTEMP

Description: New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp . (Contributed by BJ, 28-Jul-2023) The proof should not rely on df-nf except via df-nf directly. (Proof modification is discouraged.)

Ref Expression
Assertion bj-nfnnfTEMP ( Ⅎ' 𝑥 𝜑 ↔ Ⅎ 𝑥 𝜑 )

Proof

Step Hyp Ref Expression
1 bj-dfnnf3 ( Ⅎ' 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
2 df-nf ( Ⅎ 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
3 1 2 bitr4i ( Ⅎ' 𝑥 𝜑 ↔ Ⅎ 𝑥 𝜑 )