Theorem clelsb4 2143

Description: Substitution applied to an atomic wff (class version of elsb4 1853). (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 1421	. . 3 |- F/ y A e. w
2	1	sbco2 1839	. 2 |- ( [ x / y ] [ y / w ] A e. w <-> [ x / w ] A e. w )
3		nfv 1421	. . . 4 |- F/ w A e. y
4		eleq2 2101	. . . 4 |- ( w = y -> ( A e. w <-> A e. y ) )
5	3, 4	sbie 1674	. . 3 |- ( [ y / w ] A e. w <-> A e. y )
6	5	sbbii 1648	. 2 |- ( [ x / y ] [ y / w ] A e. w <-> [ x / y ] A e. y )
7		nfv 1421	. . 3 |- F/ w A e. x
8		eleq2 2101	. . 3 |- ( w = x -> ( A e. w <-> A e. x ) )
9	7, 8	sbie 1674	. 2 |- ( [ x / w ] A e. w <-> A e. x )
10	2, 6, 9	3bitr3i 199	1 |- ( [ x / y ] A e. y <-> A e. x )

Syntax hints:   <-> wb 98   e. wcel 1393   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036

This theorem is referenced by:  peano1  4317  peano2  4318