Metamath Proof Explorer

Theorem orbi1rVD

Description: Virtual deduction proof of orbi1r . 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 ) ->. ( ch \/ ph ) ).
3:2,?: e2 |- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ph \/ ch ) ).
4:1,3,?: e12 |- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ps \/ ch ) ).
5:4,?: e2 |- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ch \/ ps ) ).
6:5: |- (. ( ph <-> ps ) ->. ( ( ch \/ ph ) -> ( ch \/ ps ) ) ).
7:: |- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ch \/ ps ) ).
8:7,?: e2 |- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ps \/ ch ) ).
9:1,8,?: e12 |- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ph \/ ch ) ).
10:9,?: e2 |- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ch \/ ph ) ).
11:10: |- (. ( ph <-> ps ) ->. ( ( ch \/ ps ) -> ( ch \/ ph ) ) ).
12:6,11,?: e11 |- (. ( ph <-> ps ) ->. ( ( ch \/ ph ) <-> ( ch \/ ps ) ) ).
qed:12: |- ( ( ph <-> ps ) -> ( ( ch \/ ph ) <-> ( ch \/ ps ) ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion orbi1rVD ⊢ φ ↔ ψ → χ ∨ φ ↔ χ ∨ ψ

Proof

Step Hyp Ref Expression
1 idn1 ⊢ φ ↔ ψ → φ ↔ ψ
2 idn2 ⊢ φ ↔ ψ , χ ∨ φ → χ ∨ φ
3 pm1.4 ⊢ χ ∨ φ → φ ∨ χ
4 2 3 e2 ⊢ φ ↔ ψ , χ ∨ φ → φ ∨ χ
5 orbi1 ⊢ φ ↔ ψ → φ ∨ χ ↔ ψ ∨ χ
6 5 biimpd ⊢ φ ↔ ψ → φ ∨ χ → ψ ∨ χ
7 1 4 6 e12 ⊢ φ ↔ ψ , χ ∨ φ → ψ ∨ χ
8 pm1.4 ⊢ ψ ∨ χ → χ ∨ ψ
9 7 8 e2 ⊢ φ ↔ ψ , χ ∨ φ → χ ∨ ψ
10 9 in2 ⊢ φ ↔ ψ → χ ∨ φ → χ ∨ ψ
11 idn2 ⊢ φ ↔ ψ , χ ∨ ψ → χ ∨ ψ
12 pm1.4 ⊢ χ ∨ ψ → ψ ∨ χ
13 11 12 e2 ⊢ φ ↔ ψ , χ ∨ ψ → ψ ∨ χ
14 5 biimprd ⊢ φ ↔ ψ → ψ ∨ χ → φ ∨ χ
15 1 13 14 e12 ⊢ φ ↔ ψ , χ ∨ ψ → φ ∨ χ
16 pm1.4 ⊢ φ ∨ χ → χ ∨ φ
17 15 16 e2 ⊢ φ ↔ ψ , χ ∨ ψ → χ ∨ φ
18 17 in2 ⊢ φ ↔ ψ → χ ∨ ψ → χ ∨ φ
19 impbi ⊢ χ ∨ φ → χ ∨ ψ → χ ∨ ψ → χ ∨ φ → χ ∨ φ ↔ χ ∨ ψ
20 10 18 19 e11 ⊢ φ ↔ ψ → χ ∨ φ ↔ χ ∨ ψ
21 20 in1 ⊢ φ ↔ ψ → χ ∨ φ ↔ χ ∨ ψ