Theorem bj-indeq 10724

Description: Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)

Assertion

Ref	Expression
bj-indeq	|- ( A = B -> (Ind A <-> Ind B ) )

Proof of Theorem bj-indeq

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bj-ind 10722	. 2 |- (Ind A <-> ( (/) e. A /\ A. x e. A suc x e. A ) )
2		df-bj-ind 10722	. . 3 |- (Ind B <-> ( (/) e. B /\ A. x e. B suc x e. B ) )
3		eleq2 2142	. . . . 5 |- ( A = B -> ( (/) e. A <-> (/) e. B ) )
4	3	bicomd 139	. . . 4 |- ( A = B -> ( (/) e. B <-> (/) e. A ) )
5		eleq2 2142	. . . . . 6 |- ( A = B -> ( suc x e. A <-> suc x e. B ) )
6	5	raleqbi1dv 2557	. . . . 5 |- ( A = B -> ( A. x e. A suc x e. A <-> A. x e. B suc x e. B ) )
7	6	bicomd 139	. . . 4 |- ( A = B -> ( A. x e. B suc x e. B <-> A. x e. A suc x e. A ) )
8	4, 7	anbi12d 456	. . 3 |- ( A = B -> ( ( (/) e. B /\ A. x e. B suc x e. B ) <-> ( (/) e. A /\ A. x e. A suc x e. A ) ) )
9	2, 8	syl5rbb 191	. 2 |- ( A = B -> ( ( (/) e. A /\ A. x e. A suc x e. A ) <-> Ind B ) )
10	1, 9	syl5bb 190	1 |- ( A = B -> (Ind A <-> Ind B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   (/)c0 3251   suc csuc 4120  Ind wind 10721

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-tru 1287  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-bj-ind 10722

This theorem is referenced by:  bj-omind  10729  bj-omssind  10730  bj-ssom  10731  bj-om  10732  bj-2inf  10733