Theorem bj-indeq 16064

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 2271	. . 3 |- ( A = B -> ( (/) e. A <-> (/) e. B ) )
2		eleq2 2271	. . . 4 |- ( A = B -> ( suc x e. A <-> suc x e. B ) )
3	2	raleqbi1dv 2717	. . 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 16062	. 2 |- (Ind A <-> ( (/) e. A /\ A. x e. A suc x e. A ) )
6		df-bj-ind 16062	. 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 1373   e. wcel 2178   A.wral 2486   (/)c0 3468   suc csuc 4430  Ind wind 16061

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-bj-ind 16062

This theorem is referenced by:  bj-omind  16069  bj-omssind  16070  bj-ssom  16071  bj-om  16072  bj-2inf  16073