Theorem bdceqi 15489

Description: A class equal to a bounded one is bounded. Note the use of ax-ext 2178. See also bdceqir 15490. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bdceqi.min	|- BOUNDED A
bdceqi.maj	|- A = B

Assertion

Ref	Expression
bdceqi	|- BOUNDED B

Proof of Theorem bdceqi

Step	Hyp	Ref	Expression
1		bdceqi.min	. 2 |- BOUNDED A
2		bdceqi.maj	. . 3 |- A = B
3	2	bdceq 15488	. 2 |- (BOUNDED A <-> BOUNDED B )
4	1, 3	mpbi 145	1 |- BOUNDED B

Syntax hints:   = wceq 1364  BOUNDED wbdc 15486

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-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-bd0 15459

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192  df-bdc 15487

This theorem is referenced by:  bdceqir  15490  bds  15497  bdcuni  15522