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Theorem uniabio 5567
Description: Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
uniabio  |-  ( A. x ( ph  <->  x  =  y )  ->  U. {
x  |  ph }  =  y )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem uniabio
StepHypRef Expression
1 abbi 2588 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  <->  { x  |  ph }  =  {
x  |  x  =  y } )
21biimpi 194 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
x  |  x  =  y } )
3 df-sn 4033 . . . 4  |-  { y }  =  { x  |  x  =  y }
42, 3syl6eqr 2516 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
y } )
54unieqd 4261 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  U. {
x  |  ph }  =  U. { y } )
6 vex 3112 . . 3  |-  y  e. 
_V
76unisn 4266 . 2  |-  U. {
y }  =  y
85, 7syl6eq 2514 1  |-  ( A. x ( ph  <->  x  =  y )  ->  U. {
x  |  ph }  =  y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1393    = wceq 1395   {cab 2442   {csn 4032   U.cuni 4251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-v 3111  df-un 3476  df-sn 4033  df-pr 4035  df-uni 4252
This theorem is referenced by:  iotaval  5568  iotauni  5569
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