Theorem elsb2 2217

Description: Substitution for the second argument of the non-logical predicate in an atomic formula. See elsb1 2216 for substitution for the first argument. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
elsb2	|- ( [ y / x ] z e. x <-> z e. y )

Distinct variable group:   x, z

Proof of Theorem elsb2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1579	. . . . 5 |- ( z e. x -> A. w z e. x )
2		elequ2 2214	. . . . 5 |- ( w = x -> ( z e. w <-> z e. x ) )
3	1, 2	sbieh 1843	. . . 4 |- ( [ x / w ] z e. w <-> z e. x )
4	3	sbbii 1818	. . 3 |- ( [ y / x ] [ x / w ] z e. w <-> [ y / x ] z e. x )
5		ax-17 1579	. . . 4 |- ( z e. w -> A. x z e. w )
6	5	sbco2h 2024	. . 3 |- ( [ y / x ] [ x / w ] z e. w <-> [ y / w ] z e. w )
7	4, 6	bitr3i 186	. 2 |- ( [ y / x ] z e. x <-> [ y / w ] z e. w )
8		equsb1 1838	. . . 4 |- [ y / w ] w = y
9		elequ2 2214	. . . . 5 |- ( w = y -> ( z e. w <-> z e. y ) )
10	9	sbimi 1817	. . . 4 |- ( [ y / w ] w = y -> [ y / w ] ( z e. w <-> z e. y ) )
11	8, 10	ax-mp 5	. . 3 |- [ y / w ] ( z e. w <-> z e. y )
12		sbbi 2019	. . 3 |- ( [ y / w ] ( z e. w <-> z e. y ) <-> ( [ y / w ] z e. w <-> [ y / w ] z e. y ) )
13	11, 12	mpbi 145	. 2 |- ( [ y / w ] z e. w <-> [ y / w ] z e. y )
14		ax-17 1579	. . 3 |- ( z e. y -> A. w z e. y )
15	14	sbh 1829	. 2 |- ( [ y / w ] z e. y <-> z e. y )
16	7, 13, 15	3bitri 206	1 |- ( [ y / x ] z e. x <-> z e. y )

Syntax hints:   <-> wb 105   [wsb 1815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by: (None)