Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-sbeq Structured version   Visualization version   Unicode version

Theorem bj-sbeq 31515
Description: Distribute proper substitution through an equality relation. (See sbceqg 3775). (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-sbeq  |-  ( [ y  /  x ] A  =  B  <->  [_ y  /  x ]_ A  =  [_ y  /  x ]_ B
)

Proof of Theorem bj-sbeq
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2447 . . . . 5  |-  ( A  =  B  <->  A. z
( z  e.  A  <->  z  e.  B ) )
21sbbii 1806 . . . 4  |-  ( [ y  /  x ] A  =  B  <->  [ y  /  x ] A. z
( z  e.  A  <->  z  e.  B ) )
3 sbsbc 3273 . . . 4  |-  ( [ y  /  x ] A. z ( z  e.  A  <->  z  e.  B
)  <->  [. y  /  x ]. A. z ( z  e.  A  <->  z  e.  B ) )
4 sbcal 3319 . . . 4  |-  ( [. y  /  x ]. A. z ( z  e.  A  <->  z  e.  B
)  <->  A. z [. y  /  x ]. ( z  e.  A  <->  z  e.  B ) )
52, 3, 43bitri 275 . . 3  |-  ( [ y  /  x ] A  =  B  <->  A. z [. y  /  x ]. ( z  e.  A  <->  z  e.  B ) )
6 vex 3050 . . . . 5  |-  y  e. 
_V
7 sbcbig 3314 . . . . 5  |-  ( y  e.  _V  ->  ( [. y  /  x ]. ( z  e.  A  <->  z  e.  B )  <->  ( [. y  /  x ]. z  e.  A  <->  [. y  /  x ]. z  e.  B
) ) )
86, 7ax-mp 5 . . . 4  |-  ( [. y  /  x ]. (
z  e.  A  <->  z  e.  B )  <->  ( [. y  /  x ]. z  e.  A  <->  [. y  /  x ]. z  e.  B
) )
98albii 1693 . . 3  |-  ( A. z [. y  /  x ]. ( z  e.  A  <->  z  e.  B )  <->  A. z
( [. y  /  x ]. z  e.  A  <->  [. y  /  x ]. z  e.  B )
)
10 sbcel2 3780 . . . . 5  |-  ( [. y  /  x ]. z  e.  A  <->  z  e.  [_ y  /  x ]_ A
)
11 sbcel2 3780 . . . . 5  |-  ( [. y  /  x ]. z  e.  B  <->  z  e.  [_ y  /  x ]_ B
)
1210, 11bibi12i 317 . . . 4  |-  ( (
[. y  /  x ]. z  e.  A  <->  [. y  /  x ]. z  e.  B )  <->  ( z  e.  [_ y  /  x ]_ A  <->  z  e.  [_ y  /  x ]_ B ) )
1312albii 1693 . . 3  |-  ( A. z ( [. y  /  x ]. z  e.  A  <->  [. y  /  x ]. z  e.  B
)  <->  A. z ( z  e.  [_ y  /  x ]_ A  <->  z  e.  [_ y  /  x ]_ B ) )
145, 9, 133bitri 275 . 2  |-  ( [ y  /  x ] A  =  B  <->  A. z
( z  e.  [_ y  /  x ]_ A  <->  z  e.  [_ y  /  x ]_ B ) )
15 dfcleq 2447 . 2  |-  ( [_ y  /  x ]_ A  =  [_ y  /  x ]_ B  <->  A. z ( z  e.  [_ y  /  x ]_ A  <->  z  e.  [_ y  /  x ]_ B ) )
1614, 15bitr4i 256 1  |-  ( [ y  /  x ] A  =  B  <->  [_ y  /  x ]_ A  =  [_ y  /  x ]_ B
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188   A.wal 1444    = wceq 1446   [wsb 1799    e. wcel 1889   _Vcvv 3047   [.wsbc 3269   [_csb 3365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-in 3413  df-ss 3420  df-nul 3734
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator