Theorem imbi13 37747

Description: Join three logical equivalences to form equivalence of implications. imbi13 37747 is imbi13VD 38132 without virtual deductions and was automatically derived from imbi13VD 38132 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
imbi13	⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))))))

Proof of Theorem imbi13

Step	Hyp	Ref	Expression
1		imbi12 335	. 2 ⊢ ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜒 → 𝜏) ↔ (𝜃 → 𝜂))))
2		imbi12 335	. 2 ⊢ ((𝜑 ↔ 𝜓) → (((𝜒 → 𝜏) ↔ (𝜃 → 𝜂)) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂)))))
3	1, 2	syl9r 76	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))))))

Syntax hints:   → wi 4   ↔ wb 195

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 196

This theorem is referenced by:  trsbc  37771  trsbcVD  38135