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Metamath Proof Explorer

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