Theorem clelsb4 2184

Description: Substitution applied to an atomic wff (class version of elsb4 1894). (Contributed by Jim Kingdon, 22-Nov-2018.)

Assertion

Ref	Expression
clelsb4	|- ( [ x / y ] A e. y <-> A e. x )

Distinct variable group:   y, A

Allowed substitution hint:   A( x)

Proof of Theorem clelsb4

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1461	. . 3 |- F/ y A e. w
2	1	sbco2 1880	. 2 |- ( [ x / y ] [ y / w ] A e. w <-> [ x / w ] A e. w )
3		nfv 1461	. . . 4 |- F/ w A e. y
4		eleq2 2142	. . . 4 |- ( w = y -> ( A e. w <-> A e. y ) )
5	3, 4	sbie 1714	. . 3 |- ( [ y / w ] A e. w <-> A e. y )
6	5	sbbii 1688	. 2 |- ( [ x / y ] [ y / w ] A e. w <-> [ x / y ] A e. y )
7		nfv 1461	. . 3 |- F/ w A e. x
8		eleq2 2142	. . 3 |- ( w = x -> ( A e. w <-> A e. x ) )
9	7, 8	sbie 1714	. 2 |- ( [ x / w ] A e. w <-> A e. x )
10	2, 6, 9	3bitr3i 208	1 |- ( [ x / y ] A e. y <-> A e. x )

Syntax hints:   <-> wb 103   e. wcel 1433   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077

This theorem is referenced by:  peano1  4335  peano2  4336