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Theorem imbi13VD 41092
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 40738 is imbi13VD 41092 without virtual deductions and was automatically derived from imbi13VD 41092.
1:: (   (𝜑𝜓)   ▶   (𝜑𝜓)   )
2:: (   (𝜑𝜓)   ,   (𝜒𝜃)    ▶   (𝜒𝜃)   )
3:: (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜏 𝜂)   ▶   (𝜏𝜂)   )
4:2,3: (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜏 𝜂)   ▶   ((𝜒𝜏) ↔ (𝜃𝜂))   )
5:1,4: (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜏 𝜂)   ▶   ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂)))   )
6:5: (   (𝜑𝜓)   ,   (𝜒𝜃)    ▶   ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃 𝜂))))   )
7:6: (   (𝜑𝜓)   ▶   ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃 𝜂)))))   )
qed:7: ((𝜑𝜓) → ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃 𝜂))))))
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
imbi13VD ((𝜑𝜓) → ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂))))))

Proof of Theorem imbi13VD
StepHypRef Expression
1 idn1 40792 . . . . 5 (   (𝜑𝜓)   ▶   (𝜑𝜓)   )
2 idn2 40831 . . . . . 6 (   (𝜑𝜓)   ,   (𝜒𝜃)   ▶   (𝜒𝜃)   )
3 idn3 40833 . . . . . 6 (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜏𝜂)   ▶   (𝜏𝜂)   )
4 imbi12 348 . . . . . 6 ((𝜒𝜃) → ((𝜏𝜂) → ((𝜒𝜏) ↔ (𝜃𝜂))))
52, 3, 4e23 40973 . . . . 5 (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜏𝜂)   ▶   ((𝜒𝜏) ↔ (𝜃𝜂))   )
6 imbi12 348 . . . . 5 ((𝜑𝜓) → (((𝜒𝜏) ↔ (𝜃𝜂)) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂)))))
71, 5, 6e13 40966 . . . 4 (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜏𝜂)   ▶   ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂)))   )
87in3 40827 . . 3 (   (𝜑𝜓)   ,   (𝜒𝜃)   ▶   ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂))))   )
98in2 40823 . 2 (   (𝜑𝜓)   ▶   ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂)))))   )
109in1 40789 1 ((𝜑𝜓) → ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207
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 208  df-an 397  df-3an 1083  df-vd1 40788  df-vd2 40796  df-vd3 40808
This theorem is referenced by: (None)
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