Theorem clelsb2 2310

Description: Substitution for the second argument of the membership predicate in an atomic formula (class version of elsb2 2183). (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 1550	. . 3 |- F/ x A e. w
2	1	sbco2 1992	. 2 |- ( [ y / x ] [ x / w ] A e. w <-> [ y / w ] A e. w )
3		nfv 1550	. . . 4 |- F/ w A e. x
4		eleq2 2268	. . . 4 |- ( w = x -> ( A e. w <-> A e. x ) )
5	3, 4	sbie 1813	. . 3 |- ( [ x / w ] A e. w <-> A e. x )
6	5	sbbii 1787	. 2 |- ( [ y / x ] [ x / w ] A e. w <-> [ y / x ] A e. x )
7		nfv 1550	. . 3 |- F/ w A e. y
8		eleq2 2268	. . 3 |- ( w = y -> ( A e. w <-> A e. y ) )
9	7, 8	sbie 1813	. 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 1784   e. wcel 2175

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-cleq 2197  df-clel 2200

This theorem is referenced by:  peano1  4641  peano2  4642