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Theorem cbvopab1 4191
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 1624 . . . . 5  |-  F/ v E. y ( w  =  <. x ,  y
>.  /\  ph )
2 nfv 1624 . . . . . . 7  |-  F/ x  w  =  <. v ,  y >.
3 nfs1v 2119 . . . . . . 7  |-  F/ x [ v  /  x ] ph
42, 3nfan 1834 . . . . . 6  |-  F/ x
( w  =  <. v ,  y >.  /\  [
v  /  x ] ph )
54nfex 1853 . . . . 5  |-  F/ x E. y ( w  = 
<. v ,  y >.  /\  [ v  /  x ] ph )
6 opeq1 3898 . . . . . . . 8  |-  ( x  =  v  ->  <. x ,  y >.  =  <. v ,  y >. )
76eqeq2d 2377 . . . . . . 7  |-  ( x  =  v  ->  (
w  =  <. x ,  y >.  <->  w  =  <. v ,  y >.
) )
8 sbequ12 1931 . . . . . . 7  |-  ( x  =  v  ->  ( ph 
<->  [ v  /  x ] ph ) )
97, 8anbi12d 691 . . . . . 6  |-  ( x  =  v  ->  (
( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  =  <. v ,  y >.  /\  [
v  /  x ] ph ) ) )
109exbidv 1631 . . . . 5  |-  ( x  =  v  ->  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. y
( w  =  <. v ,  y >.  /\  [
v  /  x ] ph ) ) )
111, 5, 10cbvex 1998 . . . 4  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. v E. y
( w  =  <. v ,  y >.  /\  [
v  /  x ] ph ) )
12 nfv 1624 . . . . . . 7  |-  F/ z  w  =  <. v ,  y >.
13 cbvopab1.1 . . . . . . . 8  |-  F/ z
ph
1413nfsb 2122 . . . . . . 7  |-  F/ z [ v  /  x ] ph
1512, 14nfan 1834 . . . . . 6  |-  F/ z ( w  =  <. v ,  y >.  /\  [
v  /  x ] ph )
1615nfex 1853 . . . . 5  |-  F/ z E. y ( w  =  <. v ,  y
>.  /\  [ v  /  x ] ph )
17 nfv 1624 . . . . 5  |-  F/ v E. y ( w  =  <. z ,  y
>.  /\  ps )
18 opeq1 3898 . . . . . . . 8  |-  ( v  =  z  ->  <. v ,  y >.  =  <. z ,  y >. )
1918eqeq2d 2377 . . . . . . 7  |-  ( v  =  z  ->  (
w  =  <. v ,  y >.  <->  w  =  <. z ,  y >.
) )
20 sbequ 2073 . . . . . . . 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 2051 . . . . . . . 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 1631 . . . . 5  |-  ( v  =  z  ->  ( E. y ( w  = 
<. v ,  y >.  /\  [ v  /  x ] ph )  <->  E. y
( w  =  <. z ,  y >.  /\  ps ) ) )
2716, 17, 26cbvex 1998 . . . 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 2478 . 2  |-  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
w  |  E. z E. y ( w  = 
<. z ,  y >.  /\  ps ) }
30 df-opab 4180 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
31 df-opab 4180 . 2  |-  { <. z ,  y >.  |  ps }  =  { w  |  E. z E. y
( w  =  <. z ,  y >.  /\  ps ) }
3229, 30, 313eqtr4i 2396 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1546   F/wnf 1549    = wceq 1647   [wsb 1653   {cab 2352   <.cop 3732   {copab 4178
This theorem is referenced by:  cbvopab1v  4194  cbvmpt  4212  cbvmptf  23591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-opab 4180
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