Metamath Proof Explorer

Theorem sbcim2gVD

Description: Distribution of class substitution over a left-nested implication. Similar to sbcimg . 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. sbcim2g is sbcim2gVD without virtual deductions and was automatically derived from sbcim2gVD .

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

Ref Expression
Assertion sbcim2gVD ( 𝐴 ∈ 𝐵 → ( [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) ) )

Proof

Step Hyp Ref Expression
1 idn1 ⊢ (    𝐴 ∈ 𝐵    ▶    𝐴 ∈ 𝐵    )
2 idn2 ⊢ (    𝐴 ∈ 𝐵    ,    [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) )    ▶    [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) )    )
3 sbcimg ⊢ ( 𝐴 ∈ 𝐵 → ( [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ) ) )
4 3 biimpd ⊢ ( 𝐴 ∈ 𝐵 → ( [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ) ) )
5 1 2 4 e12 ⊢ (    𝐴 ∈ 𝐵    ,    [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) )    ▶    ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) )    )
6 sbcimg ⊢ ( 𝐴 ∈ 𝐵 → ( [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) )
7 1 6 e1a ⊢ (    𝐴 ∈ 𝐵    ▶    ( [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) )    )
8 imbi2 ⊢ ( ( [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) → ( ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) ) )
9 8 biimpcd ⊢ ( ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ) → ( ( [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) → ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) ) )
10 5 7 9 e21 ⊢ (    𝐴 ∈ 𝐵    ,    [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) )    ▶    ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) )    )
11 10 in2 ⊢ (    𝐴 ∈ 𝐵    ▶    ( [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) )    )
12 idn2 ⊢ (    𝐴 ∈ 𝐵    ,    ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) )    ▶    ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) )    )
13 biimpr ⊢ ( ( [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) → ( ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) → [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ) )
14 13 imim2d ⊢ ( ( [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) → ( ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ) ) )
15 7 12 14 e12 ⊢ (    𝐴 ∈ 𝐵    ,    ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) )    ▶    ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) )    )
16 1 3 e1a ⊢ (    𝐴 ∈ 𝐵    ▶    ( [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ) )    )
17 biimpr ⊢ ( ( [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ) ) → ( ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ) → [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) ) ) )
18 17 com12 ⊢ ( ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ) → ( ( [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] ( 𝜓 → 𝜒 ) ) ) → [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) ) ) )
19 15 16 18 e21 ⊢ (    𝐴 ∈ 𝐵    ,    ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) )    ▶    [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) )    )
20 19 in2 ⊢ (    𝐴 ∈ 𝐵    ▶    ( ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) → [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) ) )    )
21 impbi ⊢ ( ( [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) ) → ( ( ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) → [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) ) ) → ( [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) ) ) )
22 11 20 21 e11 ⊢ (    𝐴 ∈ 𝐵    ▶    ( [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) )    )
23 22 in1 ⊢ ( 𝐴 ∈ 𝐵 → ( [ 𝐴 / 𝑥 ] ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 → [ 𝐴 / 𝑥 ] 𝜒 ) ) ) )