Theorem bj-nnfbii 34078

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 34076. (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-nnfbii.1	. 2 ⊢ (𝜑 ↔ 𝜓)
2		bj-nnfbi 34076	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓))
3	1, 2	bj-mpgs 33962	1 ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)

Syntax hints:   ↔ wb 208  Ⅎ'wnnf 34074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-bj-nnf 34075

This theorem is referenced by:  bj-nnfbit  34100  bj-nnfbid  34101