Theorem imbi13VD 44845

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 44491 is imbi13VD 44845 without virtual deductions and was automatically derived from imbi13VD 44845.

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 44545	. . . . 5 ⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜑 ↔ 𝜓)   )
2		idn2 44584	. . . . . 6 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ▶   (𝜒 ↔ 𝜃)   )
3		idn3 44586	. . . . . 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 44726	. . . . 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 44719	. . . 4 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜏 ↔ 𝜂)   ▶   ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂)))   )
8	7	in3 44580	. . 3 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ▶   ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))))   )
9	8	in2 44576	. 2 ⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂)))))   )
10	9	in1 44542	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 1089  df-vd1 44541  df-vd2 44549  df-vd3 44561

This theorem is referenced by: (None)