Theorem bdceqir 15049

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 15048) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 15030). (Contributed by BJ, 3-Oct-2019.)

Hypotheses

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

Assertion

Ref	Expression
bdceqir	|- BOUNDED B

Proof of Theorem bdceqir

Step	Hyp	Ref	Expression
1		bdceqir.min	. 2 |- BOUNDED A
2		bdceqir.maj	. . 3 |- B = A
3	2	eqcomi 2193	. 2 |- A = B
4	1, 3	bdceqi 15048	1 |- BOUNDED B

Syntax hints:   = wceq 1364  BOUNDED wbdc 15045

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171  ax-bd0 15018

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185  df-bdc 15046

This theorem is referenced by:  bdcrab  15057  bdccsb  15065  bdcdif  15066  bdcun  15067  bdcin  15068  bdcnulALT  15071  bdcpw  15074  bdcsn  15075  bdcpr  15076  bdctp  15077  bdcuni  15081  bdcint  15082  bdciun  15083  bdciin  15084  bdcsuc  15085  bdcriota  15088