Theorem bdceqi 13685

Description: A class equal to a bounded one is bounded. Note the use of ax-ext 2147. See also bdceqir 13686. (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 13684	. 2 |- (BOUNDED A <-> BOUNDED B )
4	1, 3	mpbi 144	1 |- BOUNDED B

Syntax hints:   = wceq 1343  BOUNDED wbdc 13682

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-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147  ax-bd0 13655

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-clel 2161  df-bdc 13683

This theorem is referenced by:  bdceqir  13686  bds  13693  bdcuni  13718