Theorem bj-nnfbd

Description: If two formulas are equivalent for all x , then nonfreeness of x in one of them is equivalent to nonfreeness in the other, deduction form. See bj-nnfbi . (Contributed by BJ, 27-Aug-2023)

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

Proof

Step	Hyp	Ref	Expression
1		bj-nnfbd.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	alrimiv	⊢ ( 𝜑 → ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) )
3		bj-nnfbi	⊢ ( ( ( 𝜓 ↔ 𝜒 ) ∧ ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) ) → ( Ⅎ' 𝑥 𝜓 ↔ Ⅎ' 𝑥 𝜒 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( Ⅎ' 𝑥 𝜓 ↔ Ⅎ' 𝑥 𝜒 ) )