Theorem bdceqir 16740

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16739) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 16721). (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 2238	. 2 |- A = B
4	1, 3	bdceqi 16739	1 |- BOUNDED B

Syntax hints:   = wceq 1398  BOUNDED wbdc 16736

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2216  ax-bd0 16709

This theorem depends on definitions:  df-bi 117  df-cleq 2227  df-clel 2230  df-bdc 16737

This theorem is referenced by:  bdcrab  16748  bdccsb  16756  bdcdif  16757  bdcun  16758  bdcin  16759  bdcnulALT  16762  bdcpw  16765  bdcsn  16766  bdcpr  16767  bdctp  16768  bdcuni  16772  bdcint  16773  bdciun  16774  bdciin  16775  bdcsuc  16776  bdcriota  16779