Theorem bj-indeq 13646

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 2228	. . 3 |- ( A = B -> ( (/) e. A <-> (/) e. B ) )
2		eleq2 2228	. . . 4 |- ( A = B -> ( suc x e. A <-> suc x e. B ) )
3	2	raleqbi1dv 2667	. . 3 |- ( A = B -> ( A. x e. A suc x e. A <-> A. x e. B suc x e. B ) )
4	1, 3	anbi12d 465	. 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 13644	. 2 |- (Ind A <-> ( (/) e. A /\ A. x e. A suc x e. A ) )
6		df-bj-ind 13644	. 2 |- (Ind B <-> ( (/) e. B /\ A. x e. B suc x e. B ) )
7	4, 5, 6	3bitr4g 222	1 |- ( A = B -> (Ind A <-> Ind B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1342   e. wcel 2135   A.wral 2442   (/)c0 3404   suc csuc 4337  Ind wind 13643

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-bj-ind 13644

This theorem is referenced by:  bj-omind  13651  bj-omssind  13652  bj-ssom  13653  bj-om  13654  bj-2inf  13655