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

Theorem iotajust 5552
Description: Soundness justification theorem for df-iota 5553. (Contributed by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
iotajust  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. { z  |  {
x  |  ph }  =  { z } }
Distinct variable groups:    x, z    ph, z    ph, y    x, y
Allowed substitution hint:    ph( x)

Proof of Theorem iotajust
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 sneq 3969 . . . . 5  |-  ( y  =  w  ->  { y }  =  { w } )
21eqeq2d 2481 . . . 4  |-  ( y  =  w  ->  ( { x  |  ph }  =  { y }  <->  { x  |  ph }  =  {
w } ) )
32cbvabv 2595 . . 3  |-  { y  |  { x  | 
ph }  =  {
y } }  =  { w  |  {
x  |  ph }  =  { w } }
4 sneq 3969 . . . . 5  |-  ( w  =  z  ->  { w }  =  { z } )
54eqeq2d 2481 . . . 4  |-  ( w  =  z  ->  ( { x  |  ph }  =  { w }  <->  { x  |  ph }  =  {
z } ) )
65cbvabv 2595 . . 3  |-  { w  |  { x  |  ph }  =  { w } }  =  {
z  |  { x  |  ph }  =  {
z } }
73, 6eqtri 2493 . 2  |-  { y  |  { x  | 
ph }  =  {
y } }  =  { z  |  {
x  |  ph }  =  { z } }
87unieqi 4199 1  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. { z  |  {
x  |  ph }  =  { z } }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452   {cab 2457   {csn 3959   U.cuni 4190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  df-sn 3960  df-uni 4191
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator