Theorem imbi12 347

Description: Closed form of imbi12i 351. Was automatically derived from its "Virtual Deduction" version and the Metamath program "MM-PA> MINIMIZE_WITH *" command. (Contributed by Alan Sare, 18-Mar-2012.)

Assertion

Ref	Expression
imbi12	⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))))

Proof of Theorem imbi12

Step	Hyp	Ref	Expression
1		simplim 167	. . 3 ⊢ (¬ ((𝜑 ↔ 𝜓) → ¬ (𝜒 ↔ 𝜃)) → (𝜑 ↔ 𝜓))
2		simprim 166	. . 3 ⊢ (¬ ((𝜑 ↔ 𝜓) → ¬ (𝜒 ↔ 𝜃)) → (𝜒 ↔ 𝜃))
3	1, 2	imbi12d 345	. 2 ⊢ (¬ ((𝜑 ↔ 𝜓) → ¬ (𝜒 ↔ 𝜃)) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))
4	3	expi 165	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205

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

This theorem depends on definitions:  df-bi 206

This theorem is referenced by:  imbi12i  351  raleqbidvv  3338  bj-imbi12  34764  ifpbi12  41095  ifpbi13  41096  imbi13  42140  imbi13VD  42494  sbcssgVD  42503