Theorem snjust 3047

Description: Soundness justification theorem for df-sn 3049. (Contributed by Rodolfo Medina, 28-Apr-2010.) (The proof was 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

Step	Hyp	Ref	Expression
1		eqeq1 1890	. . 3 |- (x = z -> (x = A <-> z = A))
2	1	cbvabv 2420	. 2 |- {x | x = A} = {z | z = A}
3		eqeq1 1890	. . 3 |- (z = y -> (z = A <-> y = A))
4	3	cbvabv 2420	. 2 |- {z | z = A} = {y | y = A}
5	2, 4	eqtri 1908	1 |- {x | x = A} = {y | y = A}

Syntax hints:   = wceq 1298  {cab 1871

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877

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