Theorem elsb3 1893

Description: Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
elsb3	|- ( [ x / y ] y e. z <-> x e. z )

Distinct variable group:   y, z

Proof of Theorem elsb3

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1459	. . . . 5 |- ( y e. z -> A. w y e. z )
2		elequ1 1640	. . . . 5 |- ( w = y -> ( w e. z <-> y e. z ) )
3	1, 2	sbieh 1713	. . . 4 |- ( [ y / w ] w e. z <-> y e. z )
4	3	sbbii 1688	. . 3 |- ( [ x / y ] [ y / w ] w e. z <-> [ x / y ] y e. z )
5		ax-17 1459	. . . 4 |- ( w e. z -> A. y w e. z )
6	5	sbco2h 1879	. . 3 |- ( [ x / y ] [ y / w ] w e. z <-> [ x / w ] w e. z )
7	4, 6	bitr3i 184	. 2 |- ( [ x / y ] y e. z <-> [ x / w ] w e. z )
8		equsb1 1708	. . . 4 |- [ x / w ] w = x
9		elequ1 1640	. . . . 5 |- ( w = x -> ( w e. z <-> x e. z ) )
10	9	sbimi 1687	. . . 4 |- ( [ x / w ] w = x -> [ x / w ] ( w e. z <-> x e. z ) )
11	8, 10	ax-mp 7	. . 3 |- [ x / w ] ( w e. z <-> x e. z )
12		sbbi 1874	. . 3 |- ( [ x / w ] ( w e. z <-> x e. z ) <-> ( [ x / w ] w e. z <-> [ x / w ] x e. z ) )
13	11, 12	mpbi 143	. 2 |- ( [ x / w ] w e. z <-> [ x / w ] x e. z )
14		ax-17 1459	. . 3 |- ( x e. z -> A. w x e. z )
15	14	sbh 1699	. 2 |- ( [ x / w ] x e. z <-> x e. z )
16	7, 13, 15	3bitri 204	1 |- ( [ x / y ] y e. z <-> x e. z )

Syntax hints:   <-> wb 103   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  cvjust  2076