Metamath Proof Explorer

Theorem 3impexpVD

Description: Virtual deduction proof of 3impexp . 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 ) <-> ( ( ph /\ ps ) /\ ch ) )
3:1,2,?: e10 |- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ( ph /\ ps ) /\ ch ) -> th ) ).
4:3,?: e1a |- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ph /\ ps ) -> ( ch -> th ) ) ).
5:4,?: e1a |- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ph -> ( ps -> ( ch -> th ) ) ) ).
6:5: |- ( ( ( ph /\ ps /\ ch ) -> th ) -> ( ph -> ( ps -> ( ch -> th ) ) ) )
7:: |- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ph -> ( ps -> ( ch -> th ) ) ) ).
8:7,?: e1a |- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ph /\ ps ) -> ( ch -> th ) ) ).
9:8,?: e1a |- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ( ph /\ ps ) /\ ch ) -> th ) ).
10:2,9,?: e01 |- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ph /\ ps /\ ch ) -> th ) ).
11:10: |- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ph /\ ps /\ ch ) -> th ) )
qed:6,11,?: e00 |- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion 3impexpVD ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) )

Proof

Step Hyp Ref Expression
1 idn1 ⊢ (    ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )    ▶    ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )    )
2 df-3an ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
3 imbi1 ⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ) → ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) ↔ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → 𝜃 ) ) )
4 3 biimpcd ⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) → ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ) → ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → 𝜃 ) ) )
5 1 2 4 e10 ⊢ (    ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )    ▶    ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → 𝜃 )    )
6 pm3.3 ⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → 𝜃 ) → ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) ) )
7 5 6 e1a ⊢ (    ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )    ▶    ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) )    )
8 pm3.3 ⊢ ( ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) )
9 7 8 e1a ⊢ (    ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )    ▶    ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )    )
10 9 in1 ⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) )
11 idn1 ⊢ (    ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )    ▶    ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )    )
12 pm3.31 ⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) → ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) ) )
13 11 12 e1a ⊢ (    ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )    ▶    ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) )    )
14 pm3.31 ⊢ ( ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → 𝜃 ) )
15 13 14 e1a ⊢ (    ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )    ▶    ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → 𝜃 )    )
16 3 biimprd ⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ) → ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → 𝜃 ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) ) )
17 2 15 16 e01 ⊢ (    ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )    ▶    ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )    )
18 17 in1 ⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) )
19 impbi ⊢ ( ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) ) → ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) ) ) )
20 10 18 19 e00 ⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) )