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Metamath Proof Explorer

Theorem bj-nnfbii

Description: If two formulas are equivalent for all x , then nonfreeness of x in one of them is equivalent to nonfreeness in the other, inference form. See bj-nnfbi . (Contributed by BJ, 18-Nov-2023)

		Ref	Expression
	Hypothesis	bj-nnfbii.1	⊢ ( 𝜑 ↔ 𝜓 )
	Assertion	bj-nnfbii	⊢ ( Ⅎ' 𝑥 𝜑 ↔ Ⅎ' 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bj-nnfbii.1	⊢ ( 𝜑 ↔ 𝜓 )
2		bj-nnfbi	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) ) → ( Ⅎ' 𝑥 𝜑 ↔ Ⅎ' 𝑥 𝜓 ) )
3	1 2	bj-mpgs	⊢ ( Ⅎ' 𝑥 𝜑 ↔ Ⅎ' 𝑥 𝜓 )