Metamath Proof Explorer

Theorem imbi13VD

Description: Join three logical equivalences to form equivalence of implications. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. imbi13 is imbi13VD without virtual deductions and was automatically derived from imbi13VD .

1:: |- (. ( ph <-> ps ) ->. ( ph <-> ps ) ).
2:: |- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ch <-> th ) ).
3:: |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ta <-> et ) ->. ( ta <-> et ) ).
4:2,3: |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ta <-> et ) ->. ( ( ch -> ta ) <-> ( th -> et ) ) ).
5:1,4: |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ta <-> et ) ->. ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ).
6:5: |- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ).
7:6: |- (. ( ph <-> ps ) ->. ( ( ch <-> th ) -> ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ) ).
qed:7: |- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ) )
(Contributed by Alan Sare, 18-Mar-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion imbi13VD ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜒 ↔ 𝜃 ) → ( ( 𝜏 ↔ 𝜂 ) → ( ( 𝜑 → ( 𝜒 → 𝜏 ) ) ↔ ( 𝜓 → ( 𝜃 → 𝜂 ) ) ) ) ) )

Proof

Step Hyp Ref Expression
1 idn1 ⊢ (    ( 𝜑 ↔ 𝜓 )    ▶    ( 𝜑 ↔ 𝜓 )    )
2 idn2 ⊢ (    ( 𝜑 ↔ 𝜓 )    ,    ( 𝜒 ↔ 𝜃 )    ▶    ( 𝜒 ↔ 𝜃 )    )
3 idn3 ⊢ (    ( 𝜑 ↔ 𝜓 )    ,    ( 𝜒 ↔ 𝜃 )    ,    ( 𝜏 ↔ 𝜂 )    ▶    ( 𝜏 ↔ 𝜂 )    )
4 imbi12 ⊢ ( ( 𝜒 ↔ 𝜃 ) → ( ( 𝜏 ↔ 𝜂 ) → ( ( 𝜒 → 𝜏 ) ↔ ( 𝜃 → 𝜂 ) ) ) )
5 2 3 4 e23 ⊢ (    ( 𝜑 ↔ 𝜓 )    ,    ( 𝜒 ↔ 𝜃 )    ,    ( 𝜏 ↔ 𝜂 )    ▶    ( ( 𝜒 → 𝜏 ) ↔ ( 𝜃 → 𝜂 ) )    )
6 imbi12 ⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( ( 𝜒 → 𝜏 ) ↔ ( 𝜃 → 𝜂 ) ) → ( ( 𝜑 → ( 𝜒 → 𝜏 ) ) ↔ ( 𝜓 → ( 𝜃 → 𝜂 ) ) ) ) )
7 1 5 6 e13 ⊢ (    ( 𝜑 ↔ 𝜓 )    ,    ( 𝜒 ↔ 𝜃 )    ,    ( 𝜏 ↔ 𝜂 )    ▶    ( ( 𝜑 → ( 𝜒 → 𝜏 ) ) ↔ ( 𝜓 → ( 𝜃 → 𝜂 ) ) )    )
8 7 in3 ⊢ (    ( 𝜑 ↔ 𝜓 )    ,    ( 𝜒 ↔ 𝜃 )    ▶    ( ( 𝜏 ↔ 𝜂 ) → ( ( 𝜑 → ( 𝜒 → 𝜏 ) ) ↔ ( 𝜓 → ( 𝜃 → 𝜂 ) ) ) )    )
9 8 in2 ⊢ (    ( 𝜑 ↔ 𝜓 )    ▶    ( ( 𝜒 ↔ 𝜃 ) → ( ( 𝜏 ↔ 𝜂 ) → ( ( 𝜑 → ( 𝜒 → 𝜏 ) ) ↔ ( 𝜓 → ( 𝜃 → 𝜂 ) ) ) ) )    )
10 9 in1 ⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜒 ↔ 𝜃 ) → ( ( 𝜏 ↔ 𝜂 ) → ( ( 𝜑 → ( 𝜒 → 𝜏 ) ) ↔ ( 𝜓 → ( 𝜃 → 𝜂 ) ) ) ) ) )