Theorem bj-nnfbd0 37354

Description: If two formulas are equivalent, then nonfreeness of a variable in one of them is equivalent to nonfreeness in the other, deduction form. The antecedent of the conclusion is in the "strong necessity" modality of modal logic (see also bj-nnftht 37349) in order not to require sp 2219 (modal T). See bj-nnfbi 37353. (Contributed by BJ, 21-Mar-2026.)

Hypothesis

Ref	Expression
bj-nnfbd0.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
bj-nnfbd0	⊢ ((𝜑 ∧ ∀𝑥𝜑) → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))

Proof of Theorem bj-nnfbd0

Step	Hyp	Ref	Expression
1		bj-nnfbd0.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	alimi 1841	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
3		bj-nnfbi 37353	. 2 ⊢ (((𝜓 ↔ 𝜒) ∧ ∀𝑥(𝜓 ↔ 𝜒)) → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))
4	1, 2, 3	syl2an 607	1 ⊢ ((𝜑 ∧ ∀𝑥𝜑) → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568  Ⅎ'wnnf 37332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-bj-nnf 37333

This theorem is referenced by:  bj-nnfbd  37375