Theorem imbi13VD 44965

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 44612 is imbi13VD 44965 without virtual deductions and was automatically derived from imbi13VD 44965.

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.)

Assertion

Ref	Expression
imbi13VD	⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))))))

Proof of Theorem imbi13VD

Step	Hyp	Ref	Expression
1		idn1 44666	. . . . 5 ⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜑 ↔ 𝜓)   )
2		idn2 44705	. . . . . 6 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ▶   (𝜒 ↔ 𝜃)   )
3		idn3 44707	. . . . . 6 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜏 ↔ 𝜂)   ▶   (𝜏 ↔ 𝜂)   )
4		imbi12 346	. . . . . 6 ⊢ ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜒 → 𝜏) ↔ (𝜃 → 𝜂))))
5	2, 3, 4	e23 44846	. . . . 5 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜏 ↔ 𝜂)   ▶   ((𝜒 → 𝜏) ↔ (𝜃 → 𝜂))   )
6		imbi12 346	. . . . 5 ⊢ ((𝜑 ↔ 𝜓) → (((𝜒 → 𝜏) ↔ (𝜃 → 𝜂)) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂)))))
7	1, 5, 6	e13 44839	. . . 4 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜏 ↔ 𝜂)   ▶   ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂)))   )
8	7	in3 44701	. . 3 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ▶   ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))))   )
9	8	in2 44697	. 2 ⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂)))))   )
10	9	in1 44663	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))))))

Syntax hints:   → wi 4   ↔ wb 206

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 207  df-an 396  df-3an 1088  df-vd1 44662  df-vd2 44670  df-vd3 44682

This theorem is referenced by: (None)