Theorem bdceqir 15784

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 15783) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 15765). (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 2209	. 2 |- A = B
4	1, 3	bdceqi 15783	1 |- BOUNDED B

Syntax hints:   = wceq 1373  BOUNDED wbdc 15780

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-ext 2187  ax-bd0 15753

This theorem depends on definitions:  df-bi 117  df-cleq 2198  df-clel 2201  df-bdc 15781

This theorem is referenced by:  bdcrab  15792  bdccsb  15800  bdcdif  15801  bdcun  15802  bdcin  15803  bdcnulALT  15806  bdcpw  15809  bdcsn  15810  bdcpr  15811  bdctp  15812  bdcuni  15816  bdcint  15817  bdciun  15818  bdciin  15819  bdcsuc  15820  bdcriota  15823