Theorem iotajust 5095

Description: Soundness justification theorem for df-iota 5096. (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 3543	. . . . 5 |- ( y = w -> { y } = { w } )
2	1	eqeq2d 2152	. . . 4 |- ( y = w -> ( { x | ph } = { y } <-> { x | ph } = { w } ) )
3	2	cbvabv 2265	. . 3 |- { y | { x | ph } = { y } } = { w | { x | ph } = { w } }
4		sneq 3543	. . . . 5 |- ( w = z -> { w } = { z } )
5	4	eqeq2d 2152	. . . 4 |- ( w = z -> ( { x | ph } = { w } <-> { x | ph } = { z } ) )
6	5	cbvabv 2265	. . 3 |- { w | { x | ph } = { w } } = { z | { x | ph } = { z } }
7	3, 6	eqtri 2161	. 2 |- { y | { x | ph } = { y } } = { z | { x | ph } = { z } }
8	7	unieqi 3754	1 |- U. { y | { x | ph } = { y } } = U. { z | { x | ph } = { z } }

Syntax hints:   = wceq 1332   {cab 2126   {csn 3532   U.cuni 3744

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-sn 3538  df-uni 3745

This theorem is referenced by: (None)