 Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  imbi13VD Structured version   Visualization version   GIF version

Theorem imbi13VD 39424
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 39043 is imbi13VD 39424 without virtual deductions and was automatically derived from imbi13VD 39424.
 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 ((𝜑𝜓) → ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂))))))

Proof of Theorem imbi13VD
StepHypRef Expression
1 idn1 39107 . . . . 5 (   (𝜑𝜓)   ▶   (𝜑𝜓)   )
2 idn2 39155 . . . . . 6 (   (𝜑𝜓)   ,   (𝜒𝜃)   ▶   (𝜒𝜃)   )
3 idn3 39157 . . . . . 6 (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜏𝜂)   ▶   (𝜏𝜂)   )
4 imbi12 335 . . . . . 6 ((𝜒𝜃) → ((𝜏𝜂) → ((𝜒𝜏) ↔ (𝜃𝜂))))
52, 3, 4e23 39299 . . . . 5 (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜏𝜂)   ▶   ((𝜒𝜏) ↔ (𝜃𝜂))   )
6 imbi12 335 . . . . 5 ((𝜑𝜓) → (((𝜒𝜏) ↔ (𝜃𝜂)) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂)))))
71, 5, 6e13 39292 . . . 4 (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜏𝜂)   ▶   ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂)))   )
87in3 39151 . . 3 (   (𝜑𝜓)   ,   (𝜒𝜃)   ▶   ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂))))   )
98in2 39147 . 2 (   (𝜑𝜓)   ▶   ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂)))))   )
109in1 39104 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 1056  df-vd1 39103  df-vd2 39111  df-vd3 39123 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator