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Theorem cbvopab1 3903
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 1466 . . . . 5  |-  F/ v E. y ( w  =  <. x ,  y
>.  /\  ph )
2 nfv 1466 . . . . . . 7  |-  F/ x  w  =  <. v ,  y >.
3 nfs1v 1863 . . . . . . 7  |-  F/ x [ v  /  x ] ph
42, 3nfan 1502 . . . . . 6  |-  F/ x
( w  =  <. v ,  y >.  /\  [
v  /  x ] ph )
54nfex 1573 . . . . 5  |-  F/ x E. y ( w  = 
<. v ,  y >.  /\  [ v  /  x ] ph )
6 opeq1 3617 . . . . . . . 8  |-  ( x  =  v  ->  <. x ,  y >.  =  <. v ,  y >. )
76eqeq2d 2099 . . . . . . 7  |-  ( x  =  v  ->  (
w  =  <. x ,  y >.  <->  w  =  <. v ,  y >.
) )
8 sbequ12 1701 . . . . . . 7  |-  ( x  =  v  ->  ( ph 
<->  [ v  /  x ] ph ) )
97, 8anbi12d 457 . . . . . 6  |-  ( x  =  v  ->  (
( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  =  <. v ,  y >.  /\  [
v  /  x ] ph ) ) )
109exbidv 1753 . . . . 5  |-  ( x  =  v  ->  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. y
( w  =  <. v ,  y >.  /\  [
v  /  x ] ph ) ) )
111, 5, 10cbvex 1686 . . . 4  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. v E. y
( w  =  <. v ,  y >.  /\  [
v  /  x ] ph ) )
12 nfv 1466 . . . . . . 7  |-  F/ z  w  =  <. v ,  y >.
13 cbvopab1.1 . . . . . . . 8  |-  F/ z
ph
1413nfsb 1870 . . . . . . 7  |-  F/ z [ v  /  x ] ph
1512, 14nfan 1502 . . . . . 6  |-  F/ z ( w  =  <. v ,  y >.  /\  [
v  /  x ] ph )
1615nfex 1573 . . . . 5  |-  F/ z E. y ( w  =  <. v ,  y
>.  /\  [ v  /  x ] ph )
17 nfv 1466 . . . . 5  |-  F/ v E. y ( w  =  <. z ,  y
>.  /\  ps )
18 opeq1 3617 . . . . . . . 8  |-  ( v  =  z  ->  <. v ,  y >.  =  <. z ,  y >. )
1918eqeq2d 2099 . . . . . . 7  |-  ( v  =  z  ->  (
w  =  <. v ,  y >.  <->  w  =  <. z ,  y >.
) )
20 sbequ 1768 . . . . . . . 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 1721 . . . . . . . 8  |-  ( [ z  /  x ] ph 
<->  ps )
2420, 23syl6bb 194 . . . . . . 7  |-  ( v  =  z  ->  ( [ v  /  x ] ph  <->  ps ) )
2519, 24anbi12d 457 . . . . . 6  |-  ( v  =  z  ->  (
( w  =  <. v ,  y >.  /\  [
v  /  x ] ph )  <->  ( w  = 
<. z ,  y >.  /\  ps ) ) )
2625exbidv 1753 . . . . 5  |-  ( v  =  z  ->  ( E. y ( w  = 
<. v ,  y >.  /\  [ v  /  x ] ph )  <->  E. y
( w  =  <. z ,  y >.  /\  ps ) ) )
2716, 17, 26cbvex 1686 . . . 4  |-  ( E. v E. y ( w  =  <. v ,  y >.  /\  [
v  /  x ] ph )  <->  E. z E. y
( w  =  <. z ,  y >.  /\  ps ) )
2811, 27bitri 182 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. z E. y
( w  =  <. z ,  y >.  /\  ps ) )
2928abbii 2203 . 2  |-  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
w  |  E. z E. y ( w  = 
<. z ,  y >.  /\  ps ) }
30 df-opab 3892 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
31 df-opab 3892 . 2  |-  { <. z ,  y >.  |  ps }  =  { w  |  E. z E. y
( w  =  <. z ,  y >.  /\  ps ) }
3229, 30, 313eqtr4i 2118 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   F/wnf 1394   E.wex 1426   [wsb 1692   {cab 2074   <.cop 3444   {copab 3890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892
This theorem is referenced by:  cbvopab1v  3906  cbvmptf  3924  cbvmpt  3925
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