Theorem bj-indeq 15575

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 2260	. . 3 |- ( A = B -> ( (/) e. A <-> (/) e. B ) )
2		eleq2 2260	. . . 4 |- ( A = B -> ( suc x e. A <-> suc x e. B ) )
3	2	raleqbi1dv 2705	. . 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 15573	. 2 |- (Ind A <-> ( (/) e. A /\ A. x e. A suc x e. A ) )
6		df-bj-ind 15573	. 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 1364   e. wcel 2167   A.wral 2475   (/)c0 3450   suc csuc 4400  Ind wind 15572

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-bj-ind 15573

This theorem is referenced by:  bj-omind  15580  bj-omssind  15581  bj-ssom  15582  bj-om  15583  bj-2inf  15584