Theorem bj-nnfbii 37415

Description: If two formulas are equivalent, then nonfreeness of a variable in one of them is equivalent to nonfreeness in the other, inference form. See bj-nnfbi 37413. (Contributed by BJ, 18-Nov-2023.)

Hypothesis

Ref	Expression
bj-nnfbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bj-nnfbii	⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)

Proof of Theorem bj-nnfbii

Step	Hyp	Ref	Expression
1		bj-nnfbi 37413	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓))
2		bj-nnfbii.1	. 2 ⊢ (𝜑 ↔ 𝜓)
3	1, 2	bj-mpgs 37244	1 ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)

This proof depends on syntax axioms:   ↔ wb 209  Ⅎ'wnnf 37392

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-bj-nnf 37393

This theorem is used by:  bj-nnfbit  37424  bj-nnfbid  37425