Theorem bdceqi 15978

Description: A class equal to a bounded one is bounded. Note the use of ax-ext 2189. See also bdceqir 15979. (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 15977	. 2 |- (BOUNDED A <-> BOUNDED B )
4	1, 3	mpbi 145	1 |- BOUNDED B

Syntax hints:   = wceq 1373  BOUNDED wbdc 15975

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-ext 2189  ax-bd0 15948

This theorem depends on definitions:  df-bi 117  df-cleq 2200  df-clel 2203  df-bdc 15976

This theorem is referenced by:  bdceqir  15979  bds  15986  bdcuni  16011