Theorem iotajust 5159

Description: Soundness justification theorem for df-iota 5160. (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 3594	. . . . 5 |- ( y = w -> { y } = { w } )
2	1	eqeq2d 2182	. . . 4 |- ( y = w -> ( { x | ph } = { y } <-> { x | ph } = { w } ) )
3	2	cbvabv 2295	. . 3 |- { y | { x | ph } = { y } } = { w | { x | ph } = { w } }
4		sneq 3594	. . . . 5 |- ( w = z -> { w } = { z } )
5	4	eqeq2d 2182	. . . 4 |- ( w = z -> ( { x | ph } = { w } <-> { x | ph } = { z } ) )
6	5	cbvabv 2295	. . 3 |- { w | { x | ph } = { w } } = { z | { x | ph } = { z } }
7	3, 6	eqtri 2191	. 2 |- { y | { x | ph } = { y } } = { z | { x | ph } = { z } }
8	7	unieqi 3806	1 |- U. { y | { x | ph } = { y } } = U. { z | { x | ph } = { z } }

Syntax hints:   = wceq 1348   {cab 2156   {csn 3583   U.cuni 3796

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 3589  df-uni 3797

This theorem is referenced by: (None)