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 31571
Description: Distribute proper substitution through an equality relation. (See sbceqg 3777). (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 2465 . . . . 5  |-  ( A  =  B  <->  A. z
( z  e.  A  <->  z  e.  B ) )
21sbbii 1812 . . . 4  |-  ( [ y  /  x ] A  =  B  <->  [ y  /  x ] A. z
( z  e.  A  <->  z  e.  B ) )
3 sbsbc 3259 . . . 4  |-  ( [ y  /  x ] A. z ( z  e.  A  <->  z  e.  B
)  <->  [. y  /  x ]. A. z ( z  e.  A  <->  z  e.  B ) )
4 sbcal 3305 . . . 4  |-  ( [. y  /  x ]. A. z ( z  e.  A  <->  z  e.  B
)  <->  A. z [. y  /  x ]. ( z  e.  A  <->  z  e.  B ) )
52, 3, 43bitri 279 . . 3  |-  ( [ y  /  x ] A  =  B  <->  A. z [. y  /  x ]. ( z  e.  A  <->  z  e.  B ) )
6 vex 3034 . . . . 5  |-  y  e. 
_V
7 sbcbig 3300 . . . . 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 1699 . . 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 3782 . . . . 5  |-  ( [. y  /  x ]. z  e.  A  <->  z  e.  [_ y  /  x ]_ A
)
11 sbcel2 3782 . . . . 5  |-  ( [. y  /  x ]. z  e.  B  <->  z  e.  [_ y  /  x ]_ B
)
1210, 11bibi12i 322 . . . 4  |-  ( (
[. y  /  x ]. z  e.  A  <->  [. y  /  x ]. z  e.  B )  <->  ( z  e.  [_ y  /  x ]_ A  <->  z  e.  [_ y  /  x ]_ B ) )
1312albii 1699 . . 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 279 . 2  |-  ( [ y  /  x ] A  =  B  <->  A. z
( z  e.  [_ y  /  x ]_ A  <->  z  e.  [_ y  /  x ]_ B ) )
15 dfcleq 2465 . 2  |-  ( [_ y  /  x ]_ A  =  [_ y  /  x ]_ B  <->  A. z ( z  e.  [_ y  /  x ]_ A  <->  z  e.  [_ y  /  x ]_ B ) )
1614, 15bitr4i 260 1  |-  ( [ y  /  x ] A  =  B  <->  [_ y  /  x ]_ A  =  [_ y  /  x ]_ B
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   A.wal 1450    = wceq 1452   [wsb 1805    e. wcel 1904   _Vcvv 3031   [.wsbc 3255   [_csb 3349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator