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Theorem 3ornot23VD 39396
Description: Virtual deduction proof of 3ornot23 39032. 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)   ,   (𝜒𝜑 𝜓)   ▶   (𝜒𝜑𝜓)   )
3:1,?: e1a 39169 (   𝜑 ∧ ¬ 𝜓)   ▶   ¬ 𝜑   )
4:1,?: e1a 39169 (   𝜑 ∧ ¬ 𝜓)   ▶   ¬ 𝜓   )
5:3,4,?: e11 39230 (   𝜑 ∧ ¬ 𝜓)   ▶   ¬ (𝜑 𝜓)   )
6:2,?: e2 39173 (   𝜑 ∧ ¬ 𝜓)   ,   (𝜒𝜑 𝜓)   ▶   (𝜒 ∨ (𝜑𝜓))   )
7:5,6,?: e12 39268 (   𝜑 ∧ ¬ 𝜓)   ,   (𝜒𝜑 𝜓)   ▶   𝜒   )
8:7: (   𝜑 ∧ ¬ 𝜓)   ▶   ((𝜒 𝜑𝜓) → 𝜒)   )
qed:8: ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 𝜑𝜓) → 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 39107 . . . . . 6 (   𝜑 ∧ ¬ 𝜓)   ▶   𝜑 ∧ ¬ 𝜓)   )
2 simpl 472 . . . . . 6 ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜑)
31, 2e1a 39169 . . . . 5 (   𝜑 ∧ ¬ 𝜓)   ▶    ¬ 𝜑   )
4 simpr 476 . . . . . 6 ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜓)
51, 4e1a 39169 . . . . 5 (   𝜑 ∧ ¬ 𝜓)   ▶    ¬ 𝜓   )
6 ioran 510 . . . . . 6 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
76simplbi2 654 . . . . 5 𝜑 → (¬ 𝜓 → ¬ (𝜑𝜓)))
83, 5, 7e11 39230 . . . 4 (   𝜑 ∧ ¬ 𝜓)   ▶    ¬ (𝜑𝜓)   )
9 idn2 39155 . . . . 5 (   𝜑 ∧ ¬ 𝜓)   ,   (𝜒𝜑𝜓)   ▶   (𝜒𝜑𝜓)   )
10 3orass 1057 . . . . . 6 ((𝜒𝜑𝜓) ↔ (𝜒 ∨ (𝜑𝜓)))
1110biimpi 206 . . . . 5 ((𝜒𝜑𝜓) → (𝜒 ∨ (𝜑𝜓)))
129, 11e2 39173 . . . 4 (   𝜑 ∧ ¬ 𝜓)   ,   (𝜒𝜑𝜓)   ▶   (𝜒 ∨ (𝜑𝜓))   )
13 orel2 397 . . . 4 (¬ (𝜑𝜓) → ((𝜒 ∨ (𝜑𝜓)) → 𝜒))
148, 12, 13e12 39268 . . 3 (   𝜑 ∧ ¬ 𝜓)   ,   (𝜒𝜑𝜓)   ▶   𝜒   )
1514in2 39147 . 2 (   𝜑 ∧ ¬ 𝜓)   ▶   ((𝜒𝜑𝜓) → 𝜒)   )
1615in1 39104 1 ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒𝜑𝜓) → 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  w3o 1053
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-or 384  df-an 385  df-3or 1055  df-vd1 39103  df-vd2 39111
This theorem is referenced by: (None)
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