Theorem bj-clelsb3 32848

Description: Remove dependency on ax-ext 2602 (and df-cleq 2615) from clelsb3 2729. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-clelsb3	|- ( [ x / y ] y e. A <-> x e. A )

Distinct variable group:   y, A

Allowed substitution hint:   A( x)

Proof of Theorem bj-clelsb3

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . 3 |- F/ y z e. A
2	1	sbco2 2415	. 2 |- ( [ x / y ] [ y / z ] z e. A <-> [ x / z ] z e. A )
3		nfv 1843	. . . 4 |- F/ z y e. A
4		eleq1w 2684	. . . 4 |- ( z = y -> ( z e. A <-> y e. A ) )
5	3, 4	sbie 2408	. . 3 |- ( [ y / z ] z e. A <-> y e. A )
6	5	sbbii 1887	. 2 |- ( [ x / y ] [ y / z ] z e. A <-> [ x / y ] y e. A )
7		nfv 1843	. . 3 |- F/ z x e. A
8		eleq1w 2684	. . 3 |- ( z = x -> ( z e. A <-> x e. A ) )
9	7, 8	sbie 2408	. 2 |- ( [ x / z ] z e. A <-> x e. A )
10	2, 6, 9	3bitr3i 290	1 |- ( [ x / y ] y e. A <-> x e. A )

Syntax hints:   <-> wb 196   [wsb 1880   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clel 2618

This theorem is referenced by:  bj-hblem  32849