Theorem bj-indeq 13127

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 13125	. 2 |- (Ind A <-> ( (/) e. A /\ A. x e. A suc x e. A ) )
2		df-bj-ind 13125	. . 3 |- (Ind B <-> ( (/) e. B /\ A. x e. B suc x e. B ) )
3		eleq2 2203	. . . . 5 |- ( A = B -> ( (/) e. A <-> (/) e. B ) )
4	3	bicomd 140	. . . 4 |- ( A = B -> ( (/) e. B <-> (/) e. A ) )
5		eleq2 2203	. . . . . 6 |- ( A = B -> ( suc x e. A <-> suc x e. B ) )
6	5	raleqbi1dv 2634	. . . . 5 |- ( A = B -> ( A. x e. A suc x e. A <-> A. x e. B suc x e. B ) )
7	6	bicomd 140	. . . 4 |- ( A = B -> ( A. x e. B suc x e. B <-> A. x e. A suc x e. A ) )
8	4, 7	anbi12d 464	. . 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 192	. 2 |- ( A = B -> ( ( (/) e. A /\ A. x e. A suc x e. A ) <-> Ind B ) )
10	1, 9	syl5bb 191	1 |- ( A = B -> (Ind A <-> Ind B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   A.wral 2416   (/)c0 3363   suc csuc 4287  Ind wind 13124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-bj-ind 13125

This theorem is referenced by:  bj-omind  13132  bj-omssind  13133  bj-ssom  13134  bj-om  13135  bj-2inf  13136