Theorem imbi12 349

Description: Closed form of imbi12i 353. 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 168	. . 3 ⊢ (¬ ((𝜑 ↔ 𝜓) → ¬ (𝜒 ↔ 𝜃)) → (𝜑 ↔ 𝜓))
2		simprim 167	. . 3 ⊢ (¬ ((𝜑 ↔ 𝜓) → ¬ (𝜒 ↔ 𝜃)) → (𝜒 ↔ 𝜃))
3	1, 2	imbi12d 347	. 2 ⊢ (¬ ((𝜑 ↔ 𝜓) → ¬ (𝜒 ↔ 𝜃)) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))
4	3	expi 166	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by:  imbi12i  353  bj-imbi12  37235  ifpbi12  44274  ifpbi13  44275  imbi13  45289  imbi13VD  45642  sbcssgVD  45651