Theorem bj-indeq 16292

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		eleq2 2293	. . 3 |- ( A = B -> ( (/) e. A <-> (/) e. B ) )
2		eleq2 2293	. . . 4 |- ( A = B -> ( suc x e. A <-> suc x e. B ) )
3	2	raleqbi1dv 2740	. . 3 |- ( A = B -> ( A. x e. A suc x e. A <-> A. x e. B suc x e. B ) )
4	1, 3	anbi12d 473	. 2 |- ( A = B -> ( ( (/) e. A /\ A. x e. A suc x e. A ) <-> ( (/) e. B /\ A. x e. B suc x e. B ) ) )
5		df-bj-ind 16290	. 2 |- (Ind A <-> ( (/) e. A /\ A. x e. A suc x e. A ) )
6		df-bj-ind 16290	. 2 |- (Ind B <-> ( (/) e. B /\ A. x e. B suc x e. B ) )
7	4, 5, 6	3bitr4g 223	1 |- ( A = B -> (Ind A <-> Ind B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   (/)c0 3491   suc csuc 4456  Ind wind 16289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-bj-ind 16290

This theorem is referenced by:  bj-omind  16297  bj-omssind  16298  bj-ssom  16299  bj-om  16300  bj-2inf  16301