Theorem clelsb2 2299

Description: Substitution for the second argument of the membership predicate in an atomic formula (class version of elsb2 2172). (Contributed by Jim Kingdon, 22-Nov-2018.)

Assertion

Ref	Expression
clelsb2	|- ( [ y / x ] A e. x <-> A e. y )

Distinct variable group:   x, A

Allowed substitution hint:   A( y)

Proof of Theorem clelsb2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1539	. . 3 |- F/ x A e. w
2	1	sbco2 1981	. 2 |- ( [ y / x ] [ x / w ] A e. w <-> [ y / w ] A e. w )
3		nfv 1539	. . . 4 |- F/ w A e. x
4		eleq2 2257	. . . 4 |- ( w = x -> ( A e. w <-> A e. x ) )
5	3, 4	sbie 1802	. . 3 |- ( [ x / w ] A e. w <-> A e. x )
6	5	sbbii 1776	. 2 |- ( [ y / x ] [ x / w ] A e. w <-> [ y / x ] A e. x )
7		nfv 1539	. . 3 |- F/ w A e. y
8		eleq2 2257	. . 3 |- ( w = y -> ( A e. w <-> A e. y ) )
9	7, 8	sbie 1802	. 2 |- ( [ y / w ] A e. w <-> A e. y )
10	2, 6, 9	3bitr3i 210	1 |- ( [ y / x ] A e. x <-> A e. y )

Syntax hints:   <-> wb 105   [wsb 1773   e. wcel 2164

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-cleq 2186  df-clel 2189

This theorem is referenced by:  peano1  4626  peano2  4627