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Theorem sbcim2gVD 28696
Description: Distribution of class substitution over a left-nested implication. Similar to sbcimg 3162. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcim2g 28334 is sbcim2gVD 28696 without virtual deductions and was automatically derived from sbcim2gVD 28696.
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.)
Assertion
Ref Expression
sbcim2gVD  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <-> 
( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ) )

Proof of Theorem sbcim2gVD
StepHypRef Expression
1 idn1 28374 . . . . . 6  |-  (. A  e.  B  ->.  A  e.  B ).
2 idn2 28423 . . . . . 6  |-  (. A  e.  B ,. [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->.  [. A  /  x ]. ( ph  ->  ( ps  ->  ch )
) ).
3 sbcimg 3162 . . . . . . 7  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <-> 
( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) ) ) )
43biimpd 199 . . . . . 6  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch )
) ) )
51, 2, 4e12 28545 . . . . 5  |-  (. A  e.  B ,. [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->.  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch )
) ).
6 sbcimg 3162 . . . . . 6  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ps  ->  ch ) 
<->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
) )
71, 6e1_ 28437 . . . . 5  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ps 
->  ch )  <->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ).
8 imbi2 315 . . . . . 6  |-  ( (
[. A  /  x ]. ( ps  ->  ch ) 
<->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
)  ->  ( ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) )  <->  ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ) )
98biimpcd 216 . . . . 5  |-  ( (
[. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) )  ->  (
( [. A  /  x ]. ( ps  ->  ch ) 
<->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
)  ->  ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ) )
105, 7, 9e21 28551 . . . 4  |-  (. A  e.  B ,. [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->.  ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ).
1110in2 28415 . . 3  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->  ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ) ).
12 idn2 28423 . . . . . 6  |-  (. A  e.  B ,. ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->.  ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ).
13 bi2 190 . . . . . . 7  |-  ( (
[. A  /  x ]. ( ps  ->  ch ) 
<->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
)  ->  ( ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )  ->  [. A  /  x ]. ( ps  ->  ch ) ) )
1413imim2d 50 . . . . . 6  |-  ( (
[. A  /  x ]. ( ps  ->  ch ) 
<->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
)  ->  ( ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  -> 
( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) ) ) )
157, 12, 14e12 28545 . . . . 5  |-  (. A  e.  B ,. ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->.  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) ) ).
161, 3e1_ 28437 . . . . 5  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch )
) ) ).
17 bi2 190 . . . . . 6  |-  ( (
[. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <-> 
( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) ) )  -> 
( ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch )
)  ->  [. A  /  x ]. ( ph  ->  ( ps  ->  ch )
) ) )
1817com12 29 . . . . 5  |-  ( (
[. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) )  ->  (
( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <-> 
( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) ) )  ->  [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) ) ) )
1915, 16, 18e21 28551 . . . 4  |-  (. A  e.  B ,. ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->.  [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) ) ).
2019in2 28415 . . 3  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->  [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) ) ) ).
21 bi3 180 . . 3  |-  ( (
[. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->  ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) )  ->  ( (
( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->  [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) ) )  ->  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <->  ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ) ) )
2211, 20, 21e11 28498 . 2  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <->  ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ) ).
2322in1 28371 1  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <-> 
( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   [.wsbc 3121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-sbc 3122  df-vd1 28370  df-vd2 28379
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