Theorem snjust 4176

Description: Soundness justification theorem for df-sn 4178. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
snjust	|- { x | x = A } = { y | y = A }

Distinct variable groups:   x, A   y, A

Proof of Theorem snjust

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . 3 |- ( x = z -> ( x = A <-> z = A ) )
2	1	cbvabv 2747	. 2 |- { x | x = A } = { z | z = A }
3		eqeq1 2626	. . 3 |- ( z = y -> ( z = A <-> y = A ) )
4	3	cbvabv 2747	. 2 |- { z | z = A } = { y | y = A }
5	2, 4	eqtri 2644	1 |- { x | x = A } = { y | y = A }

Syntax hints:   = wceq 1483   {cab 2608

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

This theorem is referenced by: (None)