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Theorem uniabio 6235
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 2368 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  <->  { x  |  ph }  =  {
x  |  x  =  y } )
21biimpi 188 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
x  |  x  =  y } )
3 df-sn 3620 . . . 4  |-  { y }  =  { x  |  x  =  y }
42, 3syl6eqr 2308 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
y } )
54unieqd 3812 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  U. {
x  |  ph }  =  U. { y } )
6 vex 2766 . . 3  |-  y  e. 
_V
76unisn 3817 . 2  |-  U. {
y }  =  y
85, 7syl6eq 2306 1  |-  ( A. x ( ph  <->  x  =  y )  ->  U. {
x  |  ph }  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1532    = wceq 1619   {cab 2244   {csn 3614   U.cuni 3801
This theorem is referenced by:  iotaval  6236  iotauni  6237
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-rex 2524  df-v 2765  df-un 3132  df-sn 3620  df-pr 3621  df-uni 3802
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