Theorem iotajust 5850

Description: Soundness justification theorem for df-iota 5851. (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 4187	. . . . 5 |- ( y = w -> { y } = { w } )
2	1	eqeq2d 2632	. . . 4 |- ( y = w -> ( { x | ph } = { y } <-> { x | ph } = { w } ) )
3	2	cbvabv 2747	. . 3 |- { y | { x | ph } = { y } } = { w | { x | ph } = { w } }
4		sneq 4187	. . . . 5 |- ( w = z -> { w } = { z } )
5	4	eqeq2d 2632	. . . 4 |- ( w = z -> ( { x | ph } = { w } <-> { x | ph } = { z } ) )
6	5	cbvabv 2747	. . 3 |- { w | { x | ph } = { w } } = { z | { x | ph } = { z } }
7	3, 6	eqtri 2644	. 2 |- { y | { x | ph } = { y } } = { z | { x | ph } = { z } }
8	7	unieqi 4445	1 |- U. { y | { x | ph } = { y } } = U. { z | { x | ph } = { z } }

Syntax hints:   = wceq 1483   {cab 2608   {csn 4177   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-sn 4178  df-uni 4437

This theorem is referenced by: (None)