Metamath Proof Explorer

Theorem 3impexpbicomVD

Description: Virtual deduction proof of 3impexpbicom . 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 <-> ta ) ) ->. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ).
2:: |- ( ( th <-> ta ) <-> ( ta <-> th ) )
3:1,2,?: e10 |- (. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ->. ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ).
4:3,?: e1a |- (. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ->. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ).
5:4: |- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) -> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )
6:: |- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ).
7:6,?: e1a |- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ).
8:7,2,?: e10 |- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ).
9:8: |- ( ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) -> ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) )
qed:5,9,?: e00 |- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion 3impexpbicomVD ⊢φ∧ψ∧χ→θ↔τ↔φ→ψ→χ→τ↔θ

Proof

Step Hyp Ref Expression
1 idn1 ⊢φ∧ψ∧χ→θ↔τ→φ∧ψ∧χ→θ↔τ
2 bicom ⊢θ↔τ↔τ↔θ
3 imbi2 ⊢θ↔τ↔τ↔θ→φ∧ψ∧χ→θ↔τ↔φ∧ψ∧χ→τ↔θ
4 3 biimpcd ⊢φ∧ψ∧χ→θ↔τ→θ↔τ↔τ↔θ→φ∧ψ∧χ→τ↔θ
5 1 2 4 e10 ⊢φ∧ψ∧χ→θ↔τ→φ∧ψ∧χ→τ↔θ
6 3impexp ⊢φ∧ψ∧χ→τ↔θ↔φ→ψ→χ→τ↔θ
7 6 biimpi ⊢φ∧ψ∧χ→τ↔θ→φ→ψ→χ→τ↔θ
8 5 7 e1a ⊢φ∧ψ∧χ→θ↔τ→φ→ψ→χ→τ↔θ
9 8 in1 ⊢φ∧ψ∧χ→θ↔τ→φ→ψ→χ→τ↔θ
10 idn1 ⊢φ→ψ→χ→τ↔θ→φ→ψ→χ→τ↔θ
11 6 biimpri ⊢φ→ψ→χ→τ↔θ→φ∧ψ∧χ→τ↔θ
12 10 11 e1a ⊢φ→ψ→χ→τ↔θ→φ∧ψ∧χ→τ↔θ
13 3 biimprcd ⊢φ∧ψ∧χ→τ↔θ→θ↔τ↔τ↔θ→φ∧ψ∧χ→θ↔τ
14 12 2 13 e10 ⊢φ→ψ→χ→τ↔θ→φ∧ψ∧χ→θ↔τ
15 14 in1 ⊢φ→ψ→χ→τ↔θ→φ∧ψ∧χ→θ↔τ
16 impbi ⊢φ∧ψ∧χ→θ↔τ→φ→ψ→χ→τ↔θ→φ→ψ→χ→τ↔θ→φ∧ψ∧χ→θ↔τ→φ∧ψ∧χ→θ↔τ↔φ→ψ→χ→τ↔θ
17 9 15 16 e00 ⊢φ∧ψ∧χ→θ↔τ↔φ→ψ→χ→τ↔θ