Theorem bj-indeq 10053

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 10051	. 2 |- (Ind A <-> ( (/) e. A /\ A. x e. A suc x e. A ) )
2		df-bj-ind 10051	. . 3 |- (Ind B <-> ( (/) e. B /\ A. x e. B suc x e. B ) )
3		eleq2 2101	. . . . 5 |- ( A = B -> ( (/) e. A <-> (/) e. B ) )
4	3	bicomd 129	. . . 4 |- ( A = B -> ( (/) e. B <-> (/) e. A ) )
5		eleq2 2101	. . . . . 6 |- ( A = B -> ( suc x e. A <-> suc x e. B ) )
6	5	raleqbi1dv 2513	. . . . 5 |- ( A = B -> ( A. x e. A suc x e. A <-> A. x e. B suc x e. B ) )
7	6	bicomd 129	. . . 4 |- ( A = B -> ( A. x e. B suc x e. B <-> A. x e. A suc x e. A ) )
8	4, 7	anbi12d 442	. . 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 182	. 2 |- ( A = B -> ( ( (/) e. A /\ A. x e. A suc x e. A ) <-> Ind B ) )
10	1, 9	syl5bb 181	1 |- ( A = B -> (Ind A <-> Ind B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   A.wral 2306   (/)c0 3224   suc csuc 4102  Ind wind 10050

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-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-bj-ind 10051

This theorem is referenced by:  bj-omind  10058  bj-omssind  10059  bj-ssom  10060  bj-om  10061  bj-2inf  10062  peano5setOLD  10065