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Theorem cbvopab1 4281
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 1630 . . . . 5  |-  F/ v E. y ( w  =  <. x ,  y
>.  /\  ph )
2 nfv 1630 . . . . . . 7  |-  F/ x  w  =  <. v ,  y >.
3 nfs1v 2184 . . . . . . 7  |-  F/ x [ v  /  x ] ph
42, 3nfan 1847 . . . . . 6  |-  F/ x
( w  =  <. v ,  y >.  /\  [
v  /  x ] ph )
54nfex 1866 . . . . 5  |-  F/ x E. y ( w  = 
<. v ,  y >.  /\  [ v  /  x ] ph )
6 opeq1 3986 . . . . . . . 8  |-  ( x  =  v  ->  <. x ,  y >.  =  <. v ,  y >. )
76eqeq2d 2449 . . . . . . 7  |-  ( x  =  v  ->  (
w  =  <. x ,  y >.  <->  w  =  <. v ,  y >.
) )
8 sbequ12 1945 . . . . . . 7  |-  ( x  =  v  ->  ( ph 
<->  [ v  /  x ] ph ) )
97, 8anbi12d 693 . . . . . 6  |-  ( x  =  v  ->  (
( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  =  <. v ,  y >.  /\  [
v  /  x ] ph ) ) )
109exbidv 1637 . . . . 5  |-  ( x  =  v  ->  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. y
( w  =  <. v ,  y >.  /\  [
v  /  x ] ph ) ) )
111, 5, 10cbvex 1984 . . . 4  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. v E. y
( w  =  <. v ,  y >.  /\  [
v  /  x ] ph ) )
12 nfv 1630 . . . . . . 7  |-  F/ z  w  =  <. v ,  y >.
13 cbvopab1.1 . . . . . . . 8  |-  F/ z
ph
1413nfsb 2187 . . . . . . 7  |-  F/ z [ v  /  x ] ph
1512, 14nfan 1847 . . . . . 6  |-  F/ z ( w  =  <. v ,  y >.  /\  [
v  /  x ] ph )
1615nfex 1866 . . . . 5  |-  F/ z E. y ( w  =  <. v ,  y
>.  /\  [ v  /  x ] ph )
17 nfv 1630 . . . . 5  |-  F/ v E. y ( w  =  <. z ,  y
>.  /\  ps )
18 opeq1 3986 . . . . . . . 8  |-  ( v  =  z  ->  <. v ,  y >.  =  <. z ,  y >. )
1918eqeq2d 2449 . . . . . . 7  |-  ( v  =  z  ->  (
w  =  <. v ,  y >.  <->  w  =  <. z ,  y >.
) )
20 sbequ 2113 . . . . . . . 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 2152 . . . . . . . 8  |-  ( [ z  /  x ] ph 
<->  ps )
2420, 23syl6bb 254 . . . . . . 7  |-  ( v  =  z  ->  ( [ v  /  x ] ph  <->  ps ) )
2519, 24anbi12d 693 . . . . . 6  |-  ( v  =  z  ->  (
( w  =  <. v ,  y >.  /\  [
v  /  x ] ph )  <->  ( w  = 
<. z ,  y >.  /\  ps ) ) )
2625exbidv 1637 . . . . 5  |-  ( v  =  z  ->  ( E. y ( w  = 
<. v ,  y >.  /\  [ v  /  x ] ph )  <->  E. y
( w  =  <. z ,  y >.  /\  ps ) ) )
2716, 17, 26cbvex 1984 . . . 4  |-  ( E. v E. y ( w  =  <. v ,  y >.  /\  [
v  /  x ] ph )  <->  E. z E. y
( w  =  <. z ,  y >.  /\  ps ) )
2811, 27bitri 242 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. z E. y
( w  =  <. z ,  y >.  /\  ps ) )
2928abbii 2550 . 2  |-  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }  =  {
w  |  E. z E. y ( w  = 
<. z ,  y >.  /\  ps ) }
30 df-opab 4270 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
31 df-opab 4270 . 2  |-  { <. z ,  y >.  |  ps }  =  { w  |  E. z E. y
( w  =  <. z ,  y >.  /\  ps ) }
3229, 30, 313eqtr4i 2468 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551   F/wnf 1554    = wceq 1653   [wsb 1659   {cab 2424   <.cop 3819   {copab 4268
This theorem is referenced by:  cbvopab1v  4284  cbvmpt  4302  cbvmptf  24073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419
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 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4270
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