Theorem clelsb1 2455

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2103). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
clelsb1	|- ([y / x]x e. A <-> y e. A)

Distinct variable group:   x,A

Allowed substitution hint:   A(y)

Proof of Theorem clelsb1

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . 3 |- F/x w e. A
2	1	sbco2 2086	. 2 |- ([y / x][x / w]w e. A <-> [y / w]w e. A)
3		nfv 1619	. . . 4 |- F/w x e. A
4		eleq1 2413	. . . 4 |- (w = x -> (w e. A <-> x e. A))
5	3, 4	sbie 2038	. . 3 |- ([x / w]w e. A <-> x e. A)
6	5	sbbii 1653	. 2 |- ([y / x][x / w]w e. A <-> [y / x]x e. A)
7		nfv 1619	. . 3 |- F/w y e. A
8		eleq1 2413	. . 3 |- (w = y -> (w e. A <-> y e. A))
9	7, 8	sbie 2038	. 2 |- ([y / w]w e. A <-> y e. A)
10	2, 6, 9	3bitr3i 266	1 |- ([y / x]x e. A <-> y e. A)

Syntax hints:   <-> wb 176  [wsb 1648   e. wcel 1710

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

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  df-cleq 2346  df-clel 2349

This theorem is referenced by:  hblem  2458  cbvreu  2834  sbcel1gv  3106  rmo3  3134