Theorem clelsb3 1990

Description: Substitution applied to an atomic wff (class version of elsb3 1718). (Contributed by Rodolfo Medina, 28-Apr-2010.) (The proof was shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
clelsb3	|- ([x / y]y e. A <-> x e. A)

Distinct variable group:   y,A

Proof of Theorem clelsb3

Step	Hyp	Ref	Expression
1		ax-17 1317	. . . . 5 |- (y e. A -> A.w y e. A)
2		eleq1 1957	. . . . 5 |- (w = y -> (w e. A <-> y e. A))
3	1, 2	sbie 1565	. . . 4 |- ([y / w]w e. A <-> y e. A)
4	3	sbbii 1538	. . 3 |- ([x / y][y / w]w e. A <-> [x / y]y e. A)
5		ax-17 1317	. . . 4 |- (w e. A -> A.y w e. A)
6	5	sbco2 1629	. . 3 |- ([x / y][y / w]w e. A <-> [x / w]w e. A)
7	4, 6	bitr3i 192	. 2 |- ([x / y]y e. A <-> [x / w]w e. A)
8		equsb1 1561	. . . 4 |- [x / w]w = x
9		eleq1 1957	. . . . 5 |- (w = x -> (w e. A <-> x e. A))
10	9	sbimi 1537	. . . 4 |- ([x / w]w = x -> [x / w](w e. A <-> x e. A))
11	8, 10	ax-mp 7	. . 3 |- [x / w](w e. A <-> x e. A)
12		sbbi 1609	. . 3 |- ([x / w](w e. A <-> x e. A) <-> ([x / w]w e. A <-> [x / w]x e. A))
13	11, 12	mpbi 206	. 2 |- ([x / w]w e. A <-> [x / w]x e. A)
14		ax-17 1317	. . 3 |- (x e. A -> A.w x e. A)
15	14	sbf 1551	. 2 |- ([x / w]x e. A <-> x e. A)
16	7, 13, 15	3bitri 194	1 |- ([x / y]y e. A <-> x e. A)

Syntax hints:   <-> wb 163   = wceq 1298   e. wcel 1300  [wsbc 1534

This theorem is referenced by:  hblem 1993  cbvreuv 2282  sbcel1gv 2510  difjustOLD 2596  unjustOLD 2599  injustOLD 2602  sbsslemOLD 2979  bnj13 12378  bnj16 12380  bnj14OLD 12382  bnj24 12388  bnj33OLD 12402  bnj36OLD 12406  bnj45 12415  bnj62OLD 12434  bnj78 12439  bnj79OLD 12441

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-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  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-cleq 1877  df-clel 1880

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