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 ) ) ) ) )