Metamath Proof Explorer

Theorem exbiriVD

Description: Virtual deduction proof of exbiri . 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.

h1:: |- ( ( ph /\ ps ) -> ( ch <-> th ) )
2:: |- (. ph ->. ph ).
3:: |- (. ph ,. ps ->. ps ).
4:: |- (. ph ,. ps ,. th ->. th ).
5:2,1,?: e10 |- (. ph ->. ( ps -> ( ch <-> th ) ) ).
6:3,5,?: e21 |- (. ph ,. ps ->. ( ch <-> th ) ).
7:4,6,?: e32 |- (. ph ,. ps ,. th ->. ch ).
8:7: |- (. ph ,. ps ->. ( th -> ch ) ).
9:8: |- (. ph ->. ( ps -> ( th -> ch ) ) ).
qed:9: |- ( ph -> ( ps -> ( th -> ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypothesis exbiriVD.1 ⊢ φ ∧ ψ → χ ↔ θ
Assertion exbiriVD ⊢ φ → ψ → θ → χ

Proof

Step Hyp Ref Expression
1 exbiriVD.1 ⊢ φ ∧ ψ → χ ↔ θ
2 idn3 ⊢ φ , ψ , θ → θ
3 idn2 ⊢ φ , ψ → ψ
4 idn1 ⊢ φ → φ
5 pm3.3 ⊢ φ ∧ ψ → χ ↔ θ → φ → ψ → χ ↔ θ
6 5 com12 ⊢ φ → φ ∧ ψ → χ ↔ θ → ψ → χ ↔ θ
7 4 1 6 e10 ⊢ φ → ψ → χ ↔ θ
8 pm2.27 ⊢ ψ → ψ → χ ↔ θ → χ ↔ θ
9 3 7 8 e21 ⊢ φ , ψ → χ ↔ θ
10 biimpr ⊢ χ ↔ θ → θ → χ
11 10 com12 ⊢ θ → χ ↔ θ → χ
12 2 9 11 e32 ⊢ φ , ψ , θ → χ
13 12 in3 ⊢ φ , ψ → θ → χ
14 13 in2 ⊢ φ → ψ → θ → χ
15 14 in1 ⊢ φ → ψ → θ → χ