Theorem bj-hbxfrbi 37482

Description: Closed form of hbxfrbi 1858. Note: it is less important than nfbiit 1884. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 37615) in order not to require sp 2220 (modal T). See bj-hbyfrbi 37483 for its version with existential quantifiers. (Contributed by BJ, 6-May-2019.)

Assertion

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

Proof of Theorem bj-hbxfrbi

Step	Hyp	Ref	Expression
1		simpl 488	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (𝜑 ↔ 𝜓))
2		albi 1851	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
3	2	adantl 487	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
4	1, 3	imbi12d 347	1 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568

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

This theorem is used by:  bj-nnfbi  37619