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Theorem iotajust 5379
Description: Soundness justification theorem for df-iota 5380. (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 3886 . . . . 5  |-  ( y  =  w  ->  { y }  =  { w } )
21eqeq2d 2453 . . . 4  |-  ( y  =  w  ->  ( { x  |  ph }  =  { y }  <->  { x  |  ph }  =  {
w } ) )
32cbvabv 2561 . . 3  |-  { y  |  { x  | 
ph }  =  {
y } }  =  { w  |  {
x  |  ph }  =  { w } }
4 sneq 3886 . . . . 5  |-  ( w  =  z  ->  { w }  =  { z } )
54eqeq2d 2453 . . . 4  |-  ( w  =  z  ->  ( { x  |  ph }  =  { w }  <->  { x  |  ph }  =  {
z } ) )
65cbvabv 2561 . . 3  |-  { w  |  { x  |  ph }  =  { w } }  =  {
z  |  { x  |  ph }  =  {
z } }
73, 6eqtri 2462 . 2  |-  { y  |  { x  | 
ph }  =  {
y } }  =  { z  |  {
x  |  ph }  =  { z } }
87unieqi 4099 1  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. { z  |  {
x  |  ph }  =  { z } }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369   {cab 2428   {csn 3876   U.cuni 4090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rex 2720  df-sn 3877  df-uni 4091
This theorem is referenced by: (None)
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