Metamath Proof Explorer

Theorem exbirVD

Description: Virtual deduction proof of exbir . 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 ) -> ( ch <-> th ) ) ->. ( ( ph /\ ps ) -> ( ch <-> th ) ) ).
2:: |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ,. ( ph /\ ps ) ->. ( ph /\ ps ) ).
3:: |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ,. ( ph /\ ps ) , th ->. th ).
5:1,2,?: e12 |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) , ( ph /\ ps ) ->. ( ch <-> th ) ).
6:3,5,?: e32 |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) , ( ph /\ ps ) , th ->. ch ).
7:6: |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) , ( ph /\ ps ) ->. ( th -> ch ) ).
8:7: |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ->. ( ( ph /\ ps ) -> ( th -> ch ) ) ).
9:8,?: e1a |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ->. ( ph -> ( ps -> ( th -> ch ) ) ) ).
qed:9: |- ( ( ( ph /\ ps ) -> ( ch <-> th ) ) -> ( ph -> ( ps -> ( th -> ch ) ) ) )
(Contributed by Alan Sare, 13-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion exbirVD ⊢ φ ∧ ψ → χ ↔ θ → φ → ψ → θ → χ

Proof

Step Hyp Ref Expression
1 idn3 ⊢ φ ∧ ψ → χ ↔ θ , φ ∧ ψ , θ → θ
2 idn1 ⊢ φ ∧ ψ → χ ↔ θ → φ ∧ ψ → χ ↔ θ
3 idn2 ⊢ φ ∧ ψ → χ ↔ θ , φ ∧ ψ → φ ∧ ψ
4 id ⊢ φ ∧ ψ → χ ↔ θ → φ ∧ ψ → χ ↔ θ
5 2 3 4 e12 ⊢ φ ∧ ψ → χ ↔ θ , φ ∧ ψ → χ ↔ θ
6 biimpr ⊢ χ ↔ θ → θ → χ
7 6 com12 ⊢ θ → χ ↔ θ → χ
8 1 5 7 e32 ⊢ φ ∧ ψ → χ ↔ θ , φ ∧ ψ , θ → χ
9 8 in3 ⊢ φ ∧ ψ → χ ↔ θ , φ ∧ ψ → θ → χ
10 9 in2 ⊢ φ ∧ ψ → χ ↔ θ → φ ∧ ψ → θ → χ
11 pm3.3 ⊢ φ ∧ ψ → θ → χ → φ → ψ → θ → χ
12 10 11 e1a ⊢ φ ∧ ψ → χ ↔ θ → φ → ψ → θ → χ
13 12 in1 ⊢ φ ∧ ψ → χ ↔ θ → φ → ψ → θ → χ