Theorem bj-hbxfrbi 33137

Description: Closed form of hbxfrbi 1925. Notes: it is less important than nfbiit 1952; it requires sp 2226 (unlike nfbiit 1952); 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 2226	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓))
2		albi 1919	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
3	1, 2	imbi12d 336	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-12 2222

This theorem depends on definitions:  df-bi 199  df-ex 1881

This theorem is referenced by: (None)