Metamath Proof Explorer

Theorem 3ornot23VD

Description: Virtual deduction proof of 3ornot23 . 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:: |- (. ( -. ph /\ -. ps ) ->. ( -. ph /\ -. ps ) ).
2:: |- (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ( ch \/ ph \/ ps ) ).
3:1,?: e1a |- (. ( -. ph /\ -. ps ) ->. -. ph ).
4:1,?: e1a |- (. ( -. ph /\ -. ps ) ->. -. ps ).
5:3,4,?: e11 |- (. ( -. ph /\ -. ps ) ->. -. ( ph \/ ps ) ).
6:2,?: e2 |- (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ( ch \/ ( ph \/ ps ) ) ).
7:5,6,?: e12 |- (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ch ).
8:7: |- (. ( -. ph /\ -. ps ) ->. ( ( ch \/ ph \/ ps ) -> ch ) ).
qed:8: |- ( ( -. ph /\ -. ps ) -> ( ( ch \/ ph \/ ps ) -> ch ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion 3ornot23VD ⊢¬φ∧¬ψ→χ∨φ∨ψ→χ

Proof

Step Hyp Ref Expression
1 idn1 ⊢¬φ∧¬ψ→¬φ∧¬ψ
2 simpl ⊢¬φ∧¬ψ→¬φ
3 1 2 e1a ⊢¬φ∧¬ψ→¬φ
4 simpr ⊢¬φ∧¬ψ→¬ψ
5 1 4 e1a ⊢¬φ∧¬ψ→¬ψ
6 ioran ⊢¬φ∨ψ↔¬φ∧¬ψ
7 6 simplbi2 ⊢¬φ→¬ψ→¬φ∨ψ
8 3 5 7 e11 ⊢¬φ∧¬ψ→¬φ∨ψ
9 idn2 ⊢¬φ∧¬ψ,χ∨φ∨ψ→χ∨φ∨ψ
10 3orass ⊢χ∨φ∨ψ↔χ∨φ∨ψ
11 10 biimpi ⊢χ∨φ∨ψ→χ∨φ∨ψ
12 9 11 e2 ⊢¬φ∧¬ψ,χ∨φ∨ψ→χ∨φ∨ψ
13 orel2 ⊢¬φ∨ψ→χ∨φ∨ψ→χ
14 8 12 13 e12 ⊢¬φ∧¬ψ,χ∨φ∨ψ→χ
15 14 in2 ⊢¬φ∧¬ψ→χ∨φ∨ψ→χ
16 15 in1 ⊢¬φ∧¬ψ→χ∨φ∨ψ→χ