Theorem iotajust 5152

Description: Soundness justification theorem for df-iota 5153. (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 3587	. . . . 5 |- ( y = w -> { y } = { w } )
2	1	eqeq2d 2177	. . . 4 |- ( y = w -> ( { x | ph } = { y } <-> { x | ph } = { w } ) )
3	2	cbvabv 2291	. . 3 |- { y | { x | ph } = { y } } = { w | { x | ph } = { w } }
4		sneq 3587	. . . . 5 |- ( w = z -> { w } = { z } )
5	4	eqeq2d 2177	. . . 4 |- ( w = z -> ( { x | ph } = { w } <-> { x | ph } = { z } ) )
6	5	cbvabv 2291	. . 3 |- { w | { x | ph } = { w } } = { z | { x | ph } = { z } }
7	3, 6	eqtri 2186	. 2 |- { y | { x | ph } = { y } } = { z | { x | ph } = { z } }
8	7	unieqi 3799	1 |- U. { y | { x | ph } = { y } } = U. { z | { x | ph } = { z } }

Syntax hints:   = wceq 1343   {cab 2151   {csn 3576   U.cuni 3789

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-sn 3582  df-uni 3790

This theorem is referenced by: (None)