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Theorem iotajust 5218
Description: Soundness justification theorem for df-iota 5219. (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 3651 . . . . 5  |-  ( y  =  w  ->  { y }  =  { w } )
21eqeq2d 2294 . . . 4  |-  ( y  =  w  ->  ( { x  |  ph }  =  { y }  <->  { x  |  ph }  =  {
w } ) )
32cbvabv 2402 . . 3  |-  { y  |  { x  | 
ph }  =  {
y } }  =  { w  |  {
x  |  ph }  =  { w } }
4 sneq 3651 . . . . 5  |-  ( w  =  z  ->  { w }  =  { z } )
54eqeq2d 2294 . . . 4  |-  ( w  =  z  ->  ( { x  |  ph }  =  { w }  <->  { x  |  ph }  =  {
z } ) )
65cbvabv 2402 . . 3  |-  { w  |  { x  |  ph }  =  { w } }  =  {
z  |  { x  |  ph }  =  {
z } }
73, 6eqtri 2303 . 2  |-  { y  |  { x  | 
ph }  =  {
y } }  =  { z  |  {
x  |  ph }  =  { z } }
87unieqi 3837 1  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. { z  |  {
x  |  ph }  =  { z } }
Colors of variables: wff set class
Syntax hints:    = wceq 1623   {cab 2269   {csn 3640   U.cuni 3827
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-sn 3646  df-uni 3828
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