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