Theorem iotajust 4339

Description: Soundness justification theorem for df-iota 4340. (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.

Step	Hyp	Ref	Expression
1		sneq 3745	. . . . 5 |- (y = w -> {y} = {w})
2	1	eqeq2d 2364	. . . 4 |- (y = w -> ({x | ph} = {y} <-> {x | ph} = {w}))
3	2	cbvabv 2473	. . 3 |- {y | {x | ph} = {y}} = {w | {x | ph} = {w}}
4		sneq 3745	. . . . 5 |- (w = z -> {w} = {z})
5	4	eqeq2d 2364	. . . 4 |- (w = z -> ({x | ph} = {w} <-> {x | ph} = {z}))
6	5	cbvabv 2473	. . 3 |- {w | {x | ph} = {w}} = {z | {x | ph} = {z}}
7	3, 6	eqtri 2373	. 2 |- {y | {x | ph} = {y}} = {z | {x | ph} = {z}}
8	7	unieqi 3902	1 |- U.{y | {x | ph} = {y}} = U.{z | {x | ph} = {z}}

Syntax hints:   = wceq 1642  {cab 2339  {csn 3738  U.cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-sn 3742  df-uni 3893

This theorem is referenced by: (None)