Description: Join three logical equivalences to form equivalence of implications.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
imbi13 38726 is imbi13VD 39110 without virtual deductions and was automatically
derived from imbi13VD 39110.
1:: | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜑 ↔ 𝜓) )
| 2:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ (𝜒 ↔ 𝜃) )
| 3:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜏
↔ 𝜂) ▶ (𝜏 ↔ 𝜂) )
| 4:2,3: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜏
↔ 𝜂) ▶ ((𝜒 → 𝜏) ↔ (𝜃 → 𝜂)) )
| 5:1,4: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜏
↔ 𝜂) ▶ ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))) )
| 6:5: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃
→ 𝜂)))) )
| 7:6: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ↔ 𝜃)
→ ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃
→ 𝜂))))) )
| qed:7: | ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃)
→ ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃
→ 𝜂))))))
|
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is
discouraged.) (New usage is discouraged.) |