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 ) ) ) )