Theorem elsb4 1894

Description: Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
elsb4	|- ( [ x / y ] z e. y <-> z e. x )

Distinct variable group:   y, z

Proof of Theorem elsb4

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1459	. . . . 5 |- ( z e. y -> A. w z e. y )
2		elequ2 1641	. . . . 5 |- ( w = y -> ( z e. w <-> z e. y ) )
3	1, 2	sbieh 1713	. . . 4 |- ( [ y / w ] z e. w <-> z e. y )
4	3	sbbii 1688	. . 3 |- ( [ x / y ] [ y / w ] z e. w <-> [ x / y ] z e. y )
5		ax-17 1459	. . . 4 |- ( z e. w -> A. y z e. w )
6	5	sbco2h 1879	. . 3 |- ( [ x / y ] [ y / w ] z e. w <-> [ x / w ] z e. w )
7	4, 6	bitr3i 184	. 2 |- ( [ x / y ] z e. y <-> [ x / w ] z e. w )
8		equsb1 1708	. . . 4 |- [ x / w ] w = x
9		elequ2 1641	. . . . 5 |- ( w = x -> ( z e. w <-> z e. x ) )
10	9	sbimi 1687	. . . 4 |- ( [ x / w ] w = x -> [ x / w ] ( z e. w <-> z e. x ) )
11	8, 10	ax-mp 7	. . 3 |- [ x / w ] ( z e. w <-> z e. x )
12		sbbi 1874	. . 3 |- ( [ x / w ] ( z e. w <-> z e. x ) <-> ( [ x / w ] z e. w <-> [ x / w ] z e. x ) )
13	11, 12	mpbi 143	. 2 |- ( [ x / w ] z e. w <-> [ x / w ] z e. x )
14		ax-17 1459	. . 3 |- ( z e. x -> A. w z e. x )
15	14	sbh 1699	. 2 |- ( [ x / w ] z e. x <-> z e. x )
16	7, 13, 15	3bitri 204	1 |- ( [ x / y ] z e. y <-> z e. x )

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-14 1445  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:  peano2  4336