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Theorem iotajust 5376
Description: Soundness justification theorem for df-iota 5377. (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 3785 . . . . 5  |-  ( y  =  w  ->  { y }  =  { w } )
21eqeq2d 2415 . . . 4  |-  ( y  =  w  ->  ( { x  |  ph }  =  { y }  <->  { x  |  ph }  =  {
w } ) )
32cbvabv 2523 . . 3  |-  { y  |  { x  | 
ph }  =  {
y } }  =  { w  |  {
x  |  ph }  =  { w } }
4 sneq 3785 . . . . 5  |-  ( w  =  z  ->  { w }  =  { z } )
54eqeq2d 2415 . . . 4  |-  ( w  =  z  ->  ( { x  |  ph }  =  { w }  <->  { x  |  ph }  =  {
z } ) )
65cbvabv 2523 . . 3  |-  { w  |  { x  |  ph }  =  { w } }  =  {
z  |  { x  |  ph }  =  {
z } }
73, 6eqtri 2424 . 2  |-  { y  |  { x  | 
ph }  =  {
y } }  =  { z  |  {
x  |  ph }  =  { z } }
87unieqi 3985 1  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. { z  |  {
x  |  ph }  =  { z } }
Colors of variables: wff set class
Syntax hints:    = wceq 1649   {cab 2390   {csn 3774   U.cuni 3975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-sn 3780  df-uni 3976
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