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Theorem iotajust 5157
Description: Soundness justification theorem for df-iota 5158. (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 3592 . . . . 5  |-  ( y  =  w  ->  { y }  =  { w } )
21eqeq2d 2182 . . . 4  |-  ( y  =  w  ->  ( { x  |  ph }  =  { y }  <->  { x  |  ph }  =  {
w } ) )
32cbvabv 2295 . . 3  |-  { y  |  { x  | 
ph }  =  {
y } }  =  { w  |  {
x  |  ph }  =  { w } }
4 sneq 3592 . . . . 5  |-  ( w  =  z  ->  { w }  =  { z } )
54eqeq2d 2182 . . . 4  |-  ( w  =  z  ->  ( { x  |  ph }  =  { w }  <->  { x  |  ph }  =  {
z } ) )
65cbvabv 2295 . . 3  |-  { w  |  { x  |  ph }  =  { w } }  =  {
z  |  { x  |  ph }  =  {
z } }
73, 6eqtri 2191 . 2  |-  { y  |  { x  | 
ph }  =  {
y } }  =  { z  |  {
x  |  ph }  =  { z } }
87unieqi 3804 1  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. { z  |  {
x  |  ph }  =  { z } }
Colors of variables: wff set class
Syntax hints:    = wceq 1348   {cab 2156   {csn 3581   U.cuni 3794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-sn 3587  df-uni 3795
This theorem is referenced by: (None)
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