Theorem bj-nnfbi 34907

Description: If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other. Compare nfbiit 1853. From this and bj-nnfim 34928 and bj-nnfnt 34922, one can prove analogous nonfreeness conservation results for other propositional operators. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 34923) in order not to require sp 2176 (modal T). (Contributed by BJ, 27-Aug-2023.)

Assertion

Ref	Expression
bj-nnfbi	⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓))

Proof of Theorem bj-nnfbi

Step	Hyp	Ref	Expression
1		bj-hbyfrbi 34812	. . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜓 → 𝜓)))
2		bj-hbxfrbi 34811	. . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))
3	1, 2	anbi12d 631	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ↔ ((∃𝑥𝜓 → 𝜓) ∧ (𝜓 → ∀𝑥𝜓))))
4		df-bj-nnf 34906	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
5		df-bj-nnf 34906	. 2 ⊢ (Ⅎ'𝑥𝜓 ↔ ((∃𝑥𝜓 → 𝜓) ∧ (𝜓 → ∀𝑥𝜓)))
6	3, 4, 5	3bitr4g 314	1 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537  ∃wex 1782  Ⅎ'wnnf 34905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-bj-nnf 34906

This theorem is referenced by:  bj-nnfbd  34908  bj-nnfbii  34909