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Theorem exbirVD 39402
Description: Virtual deduction proof of exbir 39001. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
 1:: ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ▶   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ) 2:: ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ,    (𝜑 ∧ 𝜓)   ▶   (𝜑 ∧ 𝜓)   ) 3:: ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ,    (𝜑 ∧ 𝜓), 𝜃   ▶   𝜃   ) 5:1,2,?: e12 39268 ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)), (𝜑 ∧ 𝜓)   ▶   (𝜒 ↔ 𝜃)   ) 6:3,5,?: e32 39302 ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)), (𝜑 ∧ 𝜓), 𝜃   ▶   𝜒   ) 7:6: ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)), (𝜑 ∧ 𝜓)   ▶   (𝜃 → 𝜒)   ) 8:7: ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ▶   ((𝜑 ∧ 𝜓) → (𝜃 → 𝜒))   ) 9:8,?: e1a 39169 ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ▶   (𝜑 → (𝜓 → (𝜃 → 𝜒)))   ) qed:9: ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒))))
(Contributed by Alan Sare, 13-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
exbirVD (((𝜑𝜓) → (𝜒𝜃)) → (𝜑 → (𝜓 → (𝜃𝜒))))

Proof of Theorem exbirVD
StepHypRef Expression
1 idn3 39157 . . . . . 6 (   ((𝜑𝜓) → (𝜒𝜃))   ,   (𝜑𝜓)   ,   𝜃   ▶   𝜃   )
2 idn1 39107 . . . . . . 7 (   ((𝜑𝜓) → (𝜒𝜃))   ▶   ((𝜑𝜓) → (𝜒𝜃))   )
3 idn2 39155 . . . . . . 7 (   ((𝜑𝜓) → (𝜒𝜃))   ,   (𝜑𝜓)   ▶   (𝜑𝜓)   )
4 id 22 . . . . . . 7 (((𝜑𝜓) → (𝜒𝜃)) → ((𝜑𝜓) → (𝜒𝜃)))
52, 3, 4e12 39268 . . . . . 6 (   ((𝜑𝜓) → (𝜒𝜃))   ,   (𝜑𝜓)   ▶   (𝜒𝜃)   )
6 biimpr 210 . . . . . . 7 ((𝜒𝜃) → (𝜃𝜒))
76com12 32 . . . . . 6 (𝜃 → ((𝜒𝜃) → 𝜒))
81, 5, 7e32 39302 . . . . 5 (   ((𝜑𝜓) → (𝜒𝜃))   ,   (𝜑𝜓)   ,   𝜃   ▶   𝜒   )
98in3 39151 . . . 4 (   ((𝜑𝜓) → (𝜒𝜃))   ,   (𝜑𝜓)   ▶   (𝜃𝜒)   )
109in2 39147 . . 3 (   ((𝜑𝜓) → (𝜒𝜃))   ▶   ((𝜑𝜓) → (𝜃𝜒))   )
11 pm3.3 459 . . 3 (((𝜑𝜓) → (𝜃𝜒)) → (𝜑 → (𝜓 → (𝜃𝜒))))
1210, 11e1a 39169 . 2 (   ((𝜑𝜓) → (𝜒𝜃))   ▶   (𝜑 → (𝜓 → (𝜃𝜒)))   )
1312in1 39104 1 (((𝜑𝜓) → (𝜒𝜃)) → (𝜑 → (𝜓 → (𝜃𝜒))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383 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)
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