Theorem iotajust 5179

Description: Soundness justification theorem for df-iota 5180. (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 3605	. . . . 5 |- ( y = w -> { y } = { w } )
2	1	eqeq2d 2189	. . . 4 |- ( y = w -> ( { x | ph } = { y } <-> { x | ph } = { w } ) )
3	2	cbvabv 2302	. . 3 |- { y | { x | ph } = { y } } = { w | { x | ph } = { w } }
4		sneq 3605	. . . . 5 |- ( w = z -> { w } = { z } )
5	4	eqeq2d 2189	. . . 4 |- ( w = z -> ( { x | ph } = { w } <-> { x | ph } = { z } ) )
6	5	cbvabv 2302	. . 3 |- { w | { x | ph } = { w } } = { z | { x | ph } = { z } }
7	3, 6	eqtri 2198	. 2 |- { y | { x | ph } = { y } } = { z | { x | ph } = { z } }
8	7	unieqi 3821	1 |- U. { y | { x | ph } = { y } } = U. { z | { x | ph } = { z } }

Syntax hints:   = wceq 1353   {cab 2163   {csn 3594   U.cuni 3811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-sn 3600  df-uni 3812

This theorem is referenced by: (None)