Theorem bj-nnfbid 34082

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

Hypotheses

Ref	Expression
bj-nnfbid.1	⊢ (𝜑 → Ⅎ'𝑥𝜓)
bj-nnfbid.2	⊢ (𝜑 → Ⅎ'𝑥𝜒)

Assertion

Ref	Expression
bj-nnfbid	⊢ (𝜑 → Ⅎ'𝑥(𝜓 ↔ 𝜒))

Proof of Theorem bj-nnfbid

Step	Hyp	Ref	Expression
1		bj-nnfbid.1	. . . 4 ⊢ (𝜑 → Ⅎ'𝑥𝜓)
2		bj-nnfbid.2	. . . 4 ⊢ (𝜑 → Ⅎ'𝑥𝜒)
3		bj-nnfim 34075	. . . 4 ⊢ ((Ⅎ'𝑥𝜓 ∧ Ⅎ'𝑥𝜒) → Ⅎ'𝑥(𝜓 → 𝜒))
4	1, 2, 3	syl2anc 586	. . 3 ⊢ (𝜑 → Ⅎ'𝑥(𝜓 → 𝜒))
5		bj-nnfim 34075	. . . 4 ⊢ ((Ⅎ'𝑥𝜒 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜒 → 𝜓))
6	2, 1, 5	syl2anc 586	. . 3 ⊢ (𝜑 → Ⅎ'𝑥(𝜒 → 𝜓))
7	4, 6	bj-nnfand 34078	. 2 ⊢ (𝜑 → Ⅎ'𝑥((𝜓 → 𝜒) ∧ (𝜒 → 𝜓)))
8		dfbi2 477	. . 3 ⊢ ((𝜓 ↔ 𝜒) ↔ ((𝜓 → 𝜒) ∧ (𝜒 → 𝜓)))
9	8	bj-nnfbii 34059	. 2 ⊢ (Ⅎ'𝑥(𝜓 ↔ 𝜒) ↔ Ⅎ'𝑥((𝜓 → 𝜒) ∧ (𝜒 → 𝜓)))
10	7, 9	sylibr 236	1 ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  Ⅎ'wnnf 34055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-bj-nnf 34056

This theorem is referenced by: (None)