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Theorem uniabio 4905
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 2167 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  <->  { x  |  ph }  =  {
x  |  x  =  y } )
21biimpi 117 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
x  |  x  =  y } )
3 df-sn 3409 . . . 4  |-  { y }  =  { x  |  x  =  y }
42, 3syl6eqr 2106 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
y } )
54unieqd 3619 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  U. {
x  |  ph }  =  U. { y } )
6 vex 2577 . . 3  |-  y  e. 
_V
76unisn 3624 . 2  |-  U. {
y }  =  y
85, 7syl6eq 2104 1  |-  ( A. x ( ph  <->  x  =  y )  ->  U. {
x  |  ph }  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   A.wal 1257    = wceq 1259   {cab 2042   {csn 3403   U.cuni 3608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-uni 3609
This theorem is referenced by:  iotaval  4906  iotauni  4907
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