Theorem elsb1 2103

Description: Substitution for the first argument of the non-logical predicate in an atomic formula. See elsb2 2104 for substitution for the second argument. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
elsb1	|- ([y / x]x e. z <-> y e. z)

Distinct variable group:   x,z

Proof of Theorem elsb1

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . 3 |- F/x w e. z
2	1	sbco2 2086	. 2 |- ([y / x][x / w]w e. z <-> [y / w]w e. z)
3		nfv 1619	. . . 4 |- F/w x e. z
4		elequ1 1713	. . . 4 |- (w = x -> (w e. z <-> x e. z))
5	3, 4	sbie 2038	. . 3 |- ([x / w]w e. z <-> x e. z)
6	5	sbbii 1653	. 2 |- ([y / x][x / w]w e. z <-> [y / x]x e. z)
7		nfv 1619	. . 3 |- F/w y e. z
8		elequ1 1713	. . 3 |- (w = y -> (w e. z <-> y e. z))
9	7, 8	sbie 2038	. 2 |- ([y / w]w e. z <-> y e. z)
10	2, 6, 9	3bitr3i 266	1 |- ([y / x]x e. z <-> y e. z)

Syntax hints:   <-> wb 176  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  cvjust  2348