Theorem snjust 3581

Description: Soundness justification theorem for df-sn 3582. (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 2172	. . 3 |- ( x = z -> ( x = A <-> z = A ) )
2	1	cbvabv 2291	. 2 |- { x | x = A } = { z | z = A }
3		eqeq1 2172	. . 3 |- ( z = y -> ( z = A <-> y = A ) )
4	3	cbvabv 2291	. 2 |- { z | z = A } = { y | y = A }
5	2, 4	eqtri 2186	1 |- { x | x = A } = { y | y = A }

Syntax hints:   = wceq 1343   {cab 2151

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158

This theorem is referenced by: (None)