Theorem bj-nnfbit 34100

Description: Nonfreeness in both sides implies nonfreeness in the biconditional. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-nnfbit

Step	Hyp	Ref	Expression
1		bj-nnfim 34094	. . 3 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 → 𝜓))
2		bj-nnfim 34094	. . . 4 ⊢ ((Ⅎ'𝑥𝜓 ∧ Ⅎ'𝑥𝜑) → Ⅎ'𝑥(𝜓 → 𝜑))
3	2	ancoms 461	. . 3 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜓 → 𝜑))
4		bj-nnfan 34096	. . 3 ⊢ ((Ⅎ'𝑥(𝜑 → 𝜓) ∧ Ⅎ'𝑥(𝜓 → 𝜑)) → Ⅎ'𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
5	1, 3, 4	syl2anc 586	. 2 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
6		dfbi2 477	. . . 4 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
7	6	bicomi 226	. . 3 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ (𝜑 ↔ 𝜓))
8	7	bj-nnfbii 34078	. 2 ⊢ (Ⅎ'𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ Ⅎ'𝑥(𝜑 ↔ 𝜓))
9	5, 8	sylib 220	1 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  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: (None)