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Theorem cbvopab1 4089
Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
cbvopab1.1  |-  F/ z
ph
cbvopab1.2  |-  F/ x ps
cbvopab1.3  |-  ( x  =  z  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvopab1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  ps }
Distinct variable groups:    x, y    y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)

Proof of Theorem cbvopab1
Dummy variables  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1605 . . . . 5  |-  F/ v E. y ( w  =  <. x ,  y
>.  /\  ph )
2 nfv 1605 . . . . . . 7  |-  F/ x  w  =  <. v ,  y >.
3 nfs1v 2045 . . . . . . 7  |-  F/ x [ v  /  x ] ph
42, 3nfan 1771 . . . . . 6  |-  F/ x
( w  =  <. v ,  y >.  /\  [
v  /  x ] ph )
54nfex 1767 . . . . 5  |-  F/ x E. y ( w  = 
<. v ,  y >.  /\  [ v  /  x ] ph )
6 opeq1 3796 . . . . . . . 8  |-  ( x  =  v  ->  <. x ,  y >.  =  <. v ,  y >. )
76eqeq2d 2294 . . . . . . 7  |-  ( x  =  v  ->  (
w  =  <. x ,  y >.  <->  w  =  <. v ,  y >.
) )
8 sbequ12 1860 . . . . . . 7  |-  ( x  =  v  ->  ( ph 
<->  [ v  /  x ] ph ) )
97, 8anbi12d 691 . . . . . 6  |-  ( x  =  v  ->  (
( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  =  <. v ,  y >.  /\  [
v  /  x ] ph ) ) )
109exbidv 1612 . . . . 5  |-  ( x  =  v  ->  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. y
( w  =  <. v ,  y >.  /\  [
v  /  x ] ph ) ) )
111, 5, 10cbvex 1925 . . . 4  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. v E. y
( w  =  <. v ,  y >.  /\  [
v  /  x ] ph ) )
12 nfv 1605 . . . . . . 7  |-  F/ z  w  =  <. v ,  y >.
13 cbvopab1.1 . . . . . . . 8  |-  F/ z
ph
1413nfsb 2048 . . . . . . 7  |-  F/ z [ v  /  x ] ph
1512, 14nfan 1771 . . . . . 6  |-  F/ z ( w  =  <. v ,  y >.  /\  [
v  /  x ] ph )
1615nfex 1767 . . . . 5  |-  F/ z E. y ( w  =  <. v ,  y
>.  /\  [ v  /  x ] ph )
17 nfv 1605 . . . . 5  |-  F/ v E. y ( w  =  <. z ,  y
>.  /\  ps )
18 opeq1 3796 . . . . . . . 8  |-  ( v  =  z  ->  <. v ,  y >.  =  <. z ,  y >. )
1918eqeq2d 2294 . . . . . . 7  |-  ( v  =  z  ->  (
w  =  <. v ,  y >.  <->  w  =  <. z ,  y >.
) )
20 sbequ 2000 . . . . . . . 8  |-  ( v  =  z  ->  ( [ v  /  x ] ph  <->  [ z  /  x ] ph ) )
21 cbvopab1.2 . . . . . . . . 9  |-  F/ x ps
22 cbvopab1.3 . . . . . . . . 9  |-  ( x  =  z  ->  ( ph 
<->  ps ) )
2321, 22sbie 1978 . . . . . . . 8  |-  ( [ z  /  x ] ph 
<->  ps )
2420, 23syl6bb 252 . . . . . . 7  |-  ( v  =  z  ->  ( [ v  /  x ] ph  <->  ps ) )
2519, 24anbi12d 691 . . . . . 6  |-  ( v  =  z  ->  (
( w  =  <. v ,  y >.  /\  [
v  /  x ] ph )  <->  ( w  = 
<. z ,  y >.  /\  ps ) ) )
2625exbidv 1612 . . . . 5  |-  ( v  =  z  ->  ( E. y ( w  = 
<. v ,  y >.  /\  [ v  /  x ] ph )  <->  E. y
( w  =  <. z ,  y >.  /\  ps ) ) )
2716, 17, 26cbvex 1925 . . . 4  |-  ( E. v E. y ( w  =  <. v ,  y >.  /\  [
v  /  x ] ph )  <->  E. z E. y
( w  =  <. z ,  y >.  /\  ps ) )
2811, 27bitri 240 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. z E. y
( w  =  <. z ,  y >.  /\  ps ) )
2928abbii 2395 . 2  |-  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
w  |  E. z E. y ( w  = 
<. z ,  y >.  /\  ps ) }
30 df-opab 4078 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
31 df-opab 4078 . 2  |-  { <. z ,  y >.  |  ps }  =  { w  |  E. z E. y
( w  =  <. z ,  y >.  /\  ps ) }
3229, 30, 313eqtr4i 2313 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528   F/wnf 1531    = wceq 1623   [wsb 1629   {cab 2269   <.cop 3643   {copab 4076
This theorem is referenced by:  cbvopab1v  4092  cbvmpt  4110  cbvmptf  23220
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-3an 936  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-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078
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