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Theorem uniabio 5362
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 2491 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  <->  { x  |  ph }  =  {
x  |  x  =  y } )
21biimpi 187 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
x  |  x  =  y } )
3 df-sn 3757 . . . 4  |-  { y }  =  { x  |  x  =  y }
42, 3syl6eqr 2431 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
y } )
54unieqd 3962 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  U. {
x  |  ph }  =  U. { y } )
6 vex 2896 . . 3  |-  y  e. 
_V
76unisn 3967 . 2  |-  U. {
y }  =  y
85, 7syl6eq 2429 1  |-  ( A. x ( ph  <->  x  =  y )  ->  U. {
x  |  ph }  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    = wceq 1649   {cab 2367   {csn 3751   U.cuni 3951
This theorem is referenced by:  iotaval  5363  iotauni  5364
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-rex 2649  df-v 2895  df-un 3262  df-sn 3757  df-pr 3758  df-uni 3952
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