MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniabio Unicode version

Theorem uniabio 5229
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 2393 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  <->  { x  |  ph }  =  {
x  |  x  =  y } )
21biimpi 186 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
x  |  x  =  y } )
3 df-sn 3646 . . . 4  |-  { y }  =  { x  |  x  =  y }
42, 3syl6eqr 2333 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
y } )
54unieqd 3838 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  U. {
x  |  ph }  =  U. { y } )
6 vex 2791 . . 3  |-  y  e. 
_V
76unisn 3843 . 2  |-  U. {
y }  =  y
85, 7syl6eq 2331 1  |-  ( A. x ( ph  <->  x  =  y )  ->  U. {
x  |  ph }  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    = wceq 1623   {cab 2269   {csn 3640   U.cuni 3827
This theorem is referenced by:  iotaval  5230  iotauni  5231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790  df-un 3157  df-sn 3646  df-pr 3647  df-uni 3828
  Copyright terms: Public domain W3C validator