Theorem bdceqir 16207

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16206) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 16188). (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 2233	. 2 |- A = B
4	1, 3	bdceqi 16206	1 |- BOUNDED B

Syntax hints:   = wceq 1395  BOUNDED wbdc 16203

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211  ax-bd0 16176

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-bdc 16204

This theorem is referenced by:  bdcrab  16215  bdccsb  16223  bdcdif  16224  bdcun  16225  bdcin  16226  bdcnulALT  16229  bdcpw  16232  bdcsn  16233  bdcpr  16234  bdctp  16235  bdcuni  16239  bdcint  16240  bdciun  16241  bdciin  16242  bdcsuc  16243  bdcriota  16246