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Theorem sbcim2gVD 28651
Description: Distribution of class substitution over a left-nested implication. Similar to sbcimg 3032. 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 28302 is sbcim2gVD 28651 without virtual deductions and was automatically derived from sbcim2gVD 28651.
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 28342 . . . . . 6  |-  (. A  e.  B  ->.  A  e.  B ).
2 idn2 28385 . . . . . 6  |-  (. A  e.  B ,. [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->.  [. A  /  x ]. ( ph  ->  ( ps  ->  ch )
) ).
3 sbcimg 3032 . . . . . . 7  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <-> 
( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) ) ) )
43biimpd 198 . . . . . 6  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch )
) ) )
51, 2, 4e12 28499 . . . . 5  |-  (. A  e.  B ,. [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->.  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch )
) ).
6 sbcimg 3032 . . . . . 6  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ps  ->  ch ) 
<->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
) )
71, 6e1_ 28399 . . . . 5  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ps 
->  ch )  <->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ).
8 imbi2 314 . . . . . 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 215 . . . . 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 28505 . . . 4  |-  (. A  e.  B ,. [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->.  ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ).
1110in2 28377 . . 3  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->  ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ) ).
12 idn2 28385 . . . . . 6  |-  (. A  e.  B ,. ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->.  ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ).
13 bi2 189 . . . . . . 7  |-  ( (
[. A  /  x ]. ( ps  ->  ch ) 
<->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
)  ->  ( ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )  ->  [. A  /  x ]. ( ps  ->  ch ) ) )
1413imim2d 48 . . . . . 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 28499 . . . . 5  |-  (. A  e.  B ,. ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->.  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) ) ).
161, 3e1_ 28399 . . . . 5  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch )
) ) ).
17 bi2 189 . . . . . 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 27 . . . . 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 28505 . . . 4  |-  (. A  e.  B ,. ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->.  [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) ) ).
2019in2 28377 . . 3  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->  [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) ) ) ).
21 bi3 179 . . 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 28460 . 2  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <->  ( [. A  /  x ]. ph  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ) ).
2322in1 28339 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 176    e. wcel 1684   [.wsbc 2991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-vd1 28338  df-vd2 28347
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