Theorem imbi13VD 45454

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 45101 is imbi13VD 45454 without virtual deductions and was automatically derived from imbi13VD 45454.

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 45155	. . . . 5 ⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜑 ↔ 𝜓)   )
2		idn2 45194	. . . . . 6 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ▶   (𝜒 ↔ 𝜃)   )
3		idn3 45196	. . . . . 6 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜏 ↔ 𝜂)   ▶   (𝜏 ↔ 𝜂)   )
4		imbi12 348	. . . . . 6 ⊢ ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜒 → 𝜏) ↔ (𝜃 → 𝜂))))
5	2, 3, 4	e23 45335	. . . . 5 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜏 ↔ 𝜂)   ▶   ((𝜒 → 𝜏) ↔ (𝜃 → 𝜂))   )
6		imbi12 348	. . . . 5 ⊢ ((𝜑 ↔ 𝜓) → (((𝜒 → 𝜏) ↔ (𝜃 → 𝜂)) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂)))))
7	1, 5, 6	e13 45328	. . . 4 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜏 ↔ 𝜂)   ▶   ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂)))   )
8	7	in3 45190	. . 3 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ▶   ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))))   )
9	8	in2 45186	. 2 ⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂)))))   )
10	9	in1 45152	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))))))

Syntax hints:   → wi 4   ↔ wb 208

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 209  df-an 400  df-3an 1101  df-vd1 45151  df-vd2 45159  df-vd3 45171

This theorem is referenced by: (None)