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Metamath Proof Explorer

Theorem bj-nnfnfTEMP

Description: New nonfreeness implies old nonfreeness on minimal implicational calculus (the proof indicates it uses ax-3 because of set.mm's definition of the biconditional, but the proof actually holds in minimal implicational calculus). (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-nnfnfTEMP	⊢ ( Ⅎ' 𝑥 𝜑 → Ⅎ 𝑥 𝜑 )

Proof

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