MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvopab2v Structured version   Unicode version

Theorem cbvopab2v 4283
Description: Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)
Hypothesis
Ref Expression
cbvopab2v.1  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvopab2v  |-  { <. x ,  y >.  |  ph }  =  { <. x ,  z >.  |  ps }
Distinct variable groups:    x, y,
z    ph, z    ps, y
Allowed substitution hints:    ph( x, y)    ps( x, z)

Proof of Theorem cbvopab2v
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 opeq2 3986 . . . . . . 7  |-  ( y  =  z  ->  <. x ,  y >.  =  <. x ,  z >. )
21eqeq2d 2448 . . . . . 6  |-  ( y  =  z  ->  (
w  =  <. x ,  y >.  <->  w  =  <. x ,  z >.
) )
3 cbvopab2v.1 . . . . . 6  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
42, 3anbi12d 693 . . . . 5  |-  ( y  =  z  ->  (
( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  =  <. x ,  z >.  /\  ps ) ) )
54cbvexv 1986 . . . 4  |-  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. z
( w  =  <. x ,  z >.  /\  ps ) )
65exbii 1593 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. z
( w  =  <. x ,  z >.  /\  ps ) )
76abbii 2549 . 2  |-  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
w  |  E. x E. z ( w  = 
<. x ,  z >.  /\  ps ) }
8 df-opab 4268 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
9 df-opab 4268 . 2  |-  { <. x ,  z >.  |  ps }  =  { w  |  E. x E. z
( w  =  <. x ,  z >.  /\  ps ) }
107, 8, 93eqtr4i 2467 1  |-  { <. x ,  y >.  |  ph }  =  { <. x ,  z >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653   {cab 2423   <.cop 3818   {copab 4266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-opab 4268
  Copyright terms: Public domain W3C validator