Theorem elsb3 2434

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		nfv 1843	. . 3 |- F/ y w e. z
2	1	sbco2 2415	. 2 |- ( [ x / y ] [ y / w ] w e. z <-> [ x / w ] w e. z )
3		nfv 1843	. . . 4 |- F/ w y e. z
4		elequ1 1997	. . . 4 |- ( w = y -> ( w e. z <-> y e. z ) )
5	3, 4	sbie 2408	. . 3 |- ( [ y / w ] w e. z <-> y e. z )
6	5	sbbii 1887	. 2 |- ( [ x / y ] [ y / w ] w e. z <-> [ x / y ] y e. z )
7		nfv 1843	. . 3 |- F/ w x e. z
8		elequ1 1997	. . 3 |- ( w = x -> ( w e. z <-> x e. z ) )
9	7, 8	sbie 2408	. 2 |- ( [ x / w ] w e. z <-> x e. z )
10	2, 6, 9	3bitr3i 290	1 |- ( [ x / y ] y e. z <-> x e. z )

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-8 1992  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:  cvjust  2617