Theorem bdceqir 13736

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 13735) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 13717). (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 2169	. 2 |- A = B
4	1, 3	bdceqi 13735	1 |- BOUNDED B

Syntax hints:   = wceq 1343  BOUNDED wbdc 13732

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 13705

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-clel 2161  df-bdc 13733

This theorem is referenced by:  bdcrab  13744  bdccsb  13752  bdcdif  13753  bdcun  13754  bdcin  13755  bdcnulALT  13758  bdcpw  13761  bdcsn  13762  bdcpr  13763  bdctp  13764  bdcuni  13768  bdcint  13769  bdciun  13770  bdciin  13771  bdcsuc  13772  bdcriota  13775