Theorem elsb4 2435

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		nfv 1843	. . 3 |- F/ y z e. w
2	1	sbco2 2415	. 2 |- ( [ x / y ] [ y / w ] z e. w <-> [ x / w ] z e. w )
3		nfv 1843	. . . 4 |- F/ w z e. y
4		elequ2 2004	. . . 4 |- ( w = y -> ( z e. w <-> z e. y ) )
5	3, 4	sbie 2408	. . 3 |- ( [ y / w ] z e. w <-> z e. y )
6	5	sbbii 1887	. 2 |- ( [ x / y ] [ y / w ] z e. w <-> [ x / y ] z e. y )
7		nfv 1843	. . 3 |- F/ w z e. x
8		elequ2 2004	. . 3 |- ( w = x -> ( z e. w <-> z e. x ) )
9	7, 8	sbie 2408	. 2 |- ( [ x / w ] z e. w <-> z e. x )
10	2, 6, 9	3bitr3i 290	1 |- ( [ x / y ] z e. y <-> z e. x )

Syntax hints:   <-> wb 196   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  nfnid  4897