MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtocl4ga Structured version   Unicode version

Theorem vtocl4ga 3063
Description: Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.)
Hypotheses
Ref Expression
vtocl4ga.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtocl4ga.2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
vtocl4ga.3  |-  ( z  =  C  ->  ( ch 
<->  rh ) )
vtocl4ga.4  |-  ( w  =  D  ->  ( rh 
<->  th ) )
vtocl4ga.5  |-  ( ( ( x  e.  Q  /\  y  e.  R
)  /\  ( z  e.  S  /\  w  e.  T ) )  ->  ph )
Assertion
Ref Expression
vtocl4ga  |-  ( ( ( A  e.  Q  /\  B  e.  R
)  /\  ( C  e.  S  /\  D  e.  T ) )  ->  th )
Distinct variable groups:    x, w, y, z, A    w, B, y, z    w, C, z   
w, D    w, R, x, y, z    w, S, x, y, z    w, T, x, y, z    w, Q, x, y, z    ps, x    rh, z    th, w    ch, y
Allowed substitution hints:    ph( x, y, z, w)    ps( y,
z, w)    ch( x, z, w)    th( x, y, z)    rh( x, y, w)    B( x)    C( x, y)    D( x, y, z)

Proof of Theorem vtocl4ga
StepHypRef Expression
1 eleq1 2503 . . . . . 6  |-  ( x  =  A  ->  (
x  e.  Q  <->  A  e.  Q ) )
21anbi1d 704 . . . . 5  |-  ( x  =  A  ->  (
( x  e.  Q  /\  y  e.  R
)  <->  ( A  e.  Q  /\  y  e.  R ) ) )
32anbi1d 704 . . . 4  |-  ( x  =  A  ->  (
( ( x  e.  Q  /\  y  e.  R )  /\  (
z  e.  S  /\  w  e.  T )
)  <->  ( ( A  e.  Q  /\  y  e.  R )  /\  (
z  e.  S  /\  w  e.  T )
) ) )
4 vtocl4ga.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
53, 4imbi12d 320 . . 3  |-  ( x  =  A  ->  (
( ( ( x  e.  Q  /\  y  e.  R )  /\  (
z  e.  S  /\  w  e.  T )
)  ->  ph )  <->  ( (
( A  e.  Q  /\  y  e.  R
)  /\  ( z  e.  S  /\  w  e.  T ) )  ->  ps ) ) )
6 eleq1 2503 . . . . . 6  |-  ( y  =  B  ->  (
y  e.  R  <->  B  e.  R ) )
76anbi2d 703 . . . . 5  |-  ( y  =  B  ->  (
( A  e.  Q  /\  y  e.  R
)  <->  ( A  e.  Q  /\  B  e.  R ) ) )
87anbi1d 704 . . . 4  |-  ( y  =  B  ->  (
( ( A  e.  Q  /\  y  e.  R )  /\  (
z  e.  S  /\  w  e.  T )
)  <->  ( ( A  e.  Q  /\  B  e.  R )  /\  (
z  e.  S  /\  w  e.  T )
) ) )
9 vtocl4ga.2 . . . 4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
108, 9imbi12d 320 . . 3  |-  ( y  =  B  ->  (
( ( ( A  e.  Q  /\  y  e.  R )  /\  (
z  e.  S  /\  w  e.  T )
)  ->  ps )  <->  ( ( ( A  e.  Q  /\  B  e.  R )  /\  (
z  e.  S  /\  w  e.  T )
)  ->  ch )
) )
11 eleq1 2503 . . . . . 6  |-  ( z  =  C  ->  (
z  e.  S  <->  C  e.  S ) )
1211anbi1d 704 . . . . 5  |-  ( z  =  C  ->  (
( z  e.  S  /\  w  e.  T
)  <->  ( C  e.  S  /\  w  e.  T ) ) )
1312anbi2d 703 . . . 4  |-  ( z  =  C  ->  (
( ( A  e.  Q  /\  B  e.  R )  /\  (
z  e.  S  /\  w  e.  T )
)  <->  ( ( A  e.  Q  /\  B  e.  R )  /\  ( C  e.  S  /\  w  e.  T )
) ) )
14 vtocl4ga.3 . . . 4  |-  ( z  =  C  ->  ( ch 
<->  rh ) )
1513, 14imbi12d 320 . . 3  |-  ( z  =  C  ->  (
( ( ( A  e.  Q  /\  B  e.  R )  /\  (
z  e.  S  /\  w  e.  T )
)  ->  ch )  <->  ( ( ( A  e.  Q  /\  B  e.  R )  /\  ( C  e.  S  /\  w  e.  T )
)  ->  rh )
) )
16 eleq1 2503 . . . . . 6  |-  ( w  =  D  ->  (
w  e.  T  <->  D  e.  T ) )
1716anbi2d 703 . . . . 5  |-  ( w  =  D  ->  (
( C  e.  S  /\  w  e.  T
)  <->  ( C  e.  S  /\  D  e.  T ) ) )
1817anbi2d 703 . . . 4  |-  ( w  =  D  ->  (
( ( A  e.  Q  /\  B  e.  R )  /\  ( C  e.  S  /\  w  e.  T )
)  <->  ( ( A  e.  Q  /\  B  e.  R )  /\  ( C  e.  S  /\  D  e.  T )
) ) )
19 vtocl4ga.4 . . . 4  |-  ( w  =  D  ->  ( rh 
<->  th ) )
2018, 19imbi12d 320 . . 3  |-  ( w  =  D  ->  (
( ( ( A  e.  Q  /\  B  e.  R )  /\  ( C  e.  S  /\  w  e.  T )
)  ->  rh )  <->  ( ( ( A  e.  Q  /\  B  e.  R )  /\  ( C  e.  S  /\  D  e.  T )
)  ->  th )
) )
21 vtocl4ga.5 . . 3  |-  ( ( ( x  e.  Q  /\  y  e.  R
)  /\  ( z  e.  S  /\  w  e.  T ) )  ->  ph )
225, 10, 15, 20, 21vtocl4g 3062 . 2  |-  ( ( ( A  e.  Q  /\  B  e.  R
)  /\  ( C  e.  S  /\  D  e.  T ) )  -> 
( ( ( A  e.  Q  /\  B  e.  R )  /\  ( C  e.  S  /\  D  e.  T )
)  ->  th )
)
2322pm2.43i 47 1  |-  ( ( ( A  e.  Q  /\  B  e.  R
)  /\  ( C  e.  S  /\  D  e.  T ) )  ->  th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-v 2995
This theorem is referenced by:  wrd2ind  12393
  Copyright terms: Public domain W3C validator