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Metamath Proof Explorer

Theorem imbi12VD

Description: Implication form of imbi12i . 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. imbi12 is imbi12VD without virtual deductions and was automatically derived from imbi12VD .

1:: |- (. ( ph <-> ps ) ->. ( ph <-> ps ) ).
2:: |- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ch <-> th ) ).
3:: |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ph -> ch ) ->. ( ph -> ch ) ).
4:1,3: |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ph -> ch ) ->. ( ps -> ch ) ).
5:2,4: |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ph -> ch ) ->. ( ps -> th ) ).
6:5: |- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ph -> ch ) -> ( ps -> th ) ) ).
7:: |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ps -> th ) ->. ( ps -> th ) ).
8:1,7: |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ps -> th ) ->. ( ph -> th ) ).
9:2,8: |- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ps -> th ) ->. ( ph -> ch ) ).
10:9: |- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ps -> th ) -> ( ph -> ch ) ) ).
11:6,10: |- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ph -> ch ) <-> ( ps -> th ) ) ).
12:11: |- (. ( ph <-> ps ) ->. ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) ).
qed:12: |- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) )
(Contributed by Alan Sare, 18-Mar-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion imbi12VD ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜒 ↔ 𝜃 ) → ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜃 ) ) ) )

Proof

Step Hyp Ref Expression
1 idn2 ⊢ (    ( 𝜑 ↔ 𝜓 )    ,    ( 𝜒 ↔ 𝜃 )    ▶    ( 𝜒 ↔ 𝜃 )    )
2 idn1 ⊢ (    ( 𝜑 ↔ 𝜓 )    ▶    ( 𝜑 ↔ 𝜓 )    )
3 idn3 ⊢ (    ( 𝜑 ↔ 𝜓 )    ,    ( 𝜒 ↔ 𝜃 )    ,    ( 𝜑 → 𝜒 )    ▶    ( 𝜑 → 𝜒 )    )
4 biimpr ⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) )
5 4 imim1d ⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜑 → 𝜒 ) → ( 𝜓 → 𝜒 ) ) )
6 2 3 5 e13 ⊢ (    ( 𝜑 ↔ 𝜓 )    ,    ( 𝜒 ↔ 𝜃 )    ,    ( 𝜑 → 𝜒 )    ▶    ( 𝜓 → 𝜒 )    )
7 biimp ⊢ ( ( 𝜒 ↔ 𝜃 ) → ( 𝜒 → 𝜃 ) )
8 7 imim2d ⊢ ( ( 𝜒 ↔ 𝜃 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜓 → 𝜃 ) ) )
9 1 6 8 e23 ⊢ (    ( 𝜑 ↔ 𝜓 )    ,    ( 𝜒 ↔ 𝜃 )    ,    ( 𝜑 → 𝜒 )    ▶    ( 𝜓 → 𝜃 )    )
10 9 in3 ⊢ (    ( 𝜑 ↔ 𝜓 )    ,    ( 𝜒 ↔ 𝜃 )    ▶    ( ( 𝜑 → 𝜒 ) → ( 𝜓 → 𝜃 ) )    )
11 idn3 ⊢ (    ( 𝜑 ↔ 𝜓 )    ,    ( 𝜒 ↔ 𝜃 )    ,    ( 𝜓 → 𝜃 )    ▶    ( 𝜓 → 𝜃 )    )
12 biimp ⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 → 𝜓 ) )
13 12 imim1d ⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜓 → 𝜃 ) → ( 𝜑 → 𝜃 ) ) )
14 2 11 13 e13 ⊢ (    ( 𝜑 ↔ 𝜓 )    ,    ( 𝜒 ↔ 𝜃 )    ,    ( 𝜓 → 𝜃 )    ▶    ( 𝜑 → 𝜃 )    )
15 biimpr ⊢ ( ( 𝜒 ↔ 𝜃 ) → ( 𝜃 → 𝜒 ) )
16 15 imim2d ⊢ ( ( 𝜒 ↔ 𝜃 ) → ( ( 𝜑 → 𝜃 ) → ( 𝜑 → 𝜒 ) ) )
17 1 14 16 e23 ⊢ (    ( 𝜑 ↔ 𝜓 )    ,    ( 𝜒 ↔ 𝜃 )    ,    ( 𝜓 → 𝜃 )    ▶    ( 𝜑 → 𝜒 )    )
18 17 in3 ⊢ (    ( 𝜑 ↔ 𝜓 )    ,    ( 𝜒 ↔ 𝜃 )    ▶    ( ( 𝜓 → 𝜃 ) → ( 𝜑 → 𝜒 ) )    )
19 impbi ⊢ ( ( ( 𝜑 → 𝜒 ) → ( 𝜓 → 𝜃 ) ) → ( ( ( 𝜓 → 𝜃 ) → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜃 ) ) ) )
20 10 18 19 e22 ⊢ (    ( 𝜑 ↔ 𝜓 )    ,    ( 𝜒 ↔ 𝜃 )    ▶    ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜃 ) )    )
21 20 in2 ⊢ (    ( 𝜑 ↔ 𝜓 )    ▶    ( ( 𝜒 ↔ 𝜃 ) → ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜃 ) ) )    )
22 21 in1 ⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜒 ↔ 𝜃 ) → ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜃 ) ) ) )