Theorem bj-hbxfrbi 32608

Description: Closed form of hbxfrbi 1752. Notes: it is less important than nfbiit 1777; it requires sp 2053 (unlike nfbiit 1777); there is an obvious version with (∃𝑥𝜑 → 𝜑) instead. (Contributed by BJ, 6-May-2019.)

Assertion

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

Proof of Theorem bj-hbxfrbi

Step	Hyp	Ref	Expression
1		sp 2053	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓))
2		albi 1746	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
3	1, 2	imbi12d 334	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by: (None)