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Theorem imbi13VD 39601
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 39220 is imbi13VD 39601 without virtual deductions and was automatically derived from imbi13VD 39601.
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 39284 . . . . 5 (   (𝜑𝜓)   ▶   (𝜑𝜓)   )
2 idn2 39332 . . . . . 6 (   (𝜑𝜓)   ,   (𝜒𝜃)   ▶   (𝜒𝜃)   )
3 idn3 39334 . . . . . 6 (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜏𝜂)   ▶   (𝜏𝜂)   )
4 imbi12 335 . . . . . 6 ((𝜒𝜃) → ((𝜏𝜂) → ((𝜒𝜏) ↔ (𝜃𝜂))))
52, 3, 4e23 39476 . . . . 5 (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜏𝜂)   ▶   ((𝜒𝜏) ↔ (𝜃𝜂))   )
6 imbi12 335 . . . . 5 ((𝜑𝜓) → (((𝜒𝜏) ↔ (𝜃𝜂)) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂)))))
71, 5, 6e13 39469 . . . 4 (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜏𝜂)   ▶   ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂)))   )
87in3 39328 . . 3 (   (𝜑𝜓)   ,   (𝜒𝜃)   ▶   ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂))))   )
98in2 39324 . 2 (   (𝜑𝜓)   ▶   ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂)))))   )
109in1 39281 1 ((𝜑𝜓) → ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196
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 197  df-an 385  df-3an 1074  df-vd1 39280  df-vd2 39288  df-vd3 39300
This theorem is referenced by: (None)
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