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Theorem 3ornot23VD 41961
Description: Virtual deduction proof of 3ornot23 41623. 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)   ▶   (¬ 𝜑 ∧ ¬ 𝜓)   ) 2:: ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ,   (𝜒 ∨ 𝜑 ∨ 𝜓)   ▶   (𝜒 ∨ 𝜑 ∨ 𝜓)   ) 3:1,?: e1a 41741 ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   ¬ 𝜑   ) 4:1,?: e1a 41741 ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   ¬ 𝜓   ) 5:3,4,?: e11 41802 ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   ¬ (𝜑 ∨ 𝜓)   ) 6:2,?: e2 41745 ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ,   (𝜒 ∨ 𝜑 ∨ 𝜓)   ▶   (𝜒 ∨ (𝜑 ∨ 𝜓))   ) 7:5,6,?: e12 41838 ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ,   (𝜒 ∨ 𝜑 ∨ 𝜓)   ▶   𝜒   ) 8:7: ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒)   ) qed:8: ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
3ornot23VD ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒𝜑𝜓) → 𝜒))

Proof of Theorem 3ornot23VD
StepHypRef Expression
1 idn1 41688 . . . . . 6 (   𝜑 ∧ ¬ 𝜓)   ▶   𝜑 ∧ ¬ 𝜓)   )
2 simpl 486 . . . . . 6 ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜑)
31, 2e1a 41741 . . . . 5 (   𝜑 ∧ ¬ 𝜓)   ▶    ¬ 𝜑   )
4 simpr 488 . . . . . 6 ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜓)
51, 4e1a 41741 . . . . 5 (   𝜑 ∧ ¬ 𝜓)   ▶    ¬ 𝜓   )
6 ioran 981 . . . . . 6 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
76simplbi2 504 . . . . 5 𝜑 → (¬ 𝜓 → ¬ (𝜑𝜓)))
83, 5, 7e11 41802 . . . 4 (   𝜑 ∧ ¬ 𝜓)   ▶    ¬ (𝜑𝜓)   )
9 idn2 41727 . . . . 5 (   𝜑 ∧ ¬ 𝜓)   ,   (𝜒𝜑𝜓)   ▶   (𝜒𝜑𝜓)   )
10 3orass 1087 . . . . . 6 ((𝜒𝜑𝜓) ↔ (𝜒 ∨ (𝜑𝜓)))
1110biimpi 219 . . . . 5 ((𝜒𝜑𝜓) → (𝜒 ∨ (𝜑𝜓)))
129, 11e2 41745 . . . 4 (   𝜑 ∧ ¬ 𝜓)   ,   (𝜒𝜑𝜓)   ▶   (𝜒 ∨ (𝜑𝜓))   )
13 orel2 888 . . . 4 (¬ (𝜑𝜓) → ((𝜒 ∨ (𝜑𝜓)) → 𝜒))
148, 12, 13e12 41838 . . 3 (   𝜑 ∧ ¬ 𝜓)   ,   (𝜒𝜑𝜓)   ▶   𝜒   )
1514in2 41719 . 2 (   𝜑 ∧ ¬ 𝜓)   ▶   ((𝜒𝜑𝜓) → 𝜒)   )
1615in1 41685 1 ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒𝜑𝜓) → 𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   ∨ w3o 1083 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 210  df-an 400  df-or 845  df-3or 1085  df-vd1 41684  df-vd2 41692 This theorem is referenced by: (None)
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