Theorem bdceqir 16755

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16754) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 16736). (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 16754	1 |- BOUNDED B

Syntax hints:   = wceq 1398  BOUNDED wbdc 16751

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 16724

This theorem depends on definitions:  df-bi 117  df-cleq 2227  df-clel 2230  df-bdc 16752

This theorem is referenced by:  bdcrab  16763  bdccsb  16771  bdcdif  16772  bdcun  16773  bdcin  16774  bdcnulALT  16777  bdcpw  16780  bdcsn  16781  bdcpr  16782  bdctp  16783  bdcuni  16787  bdcint  16788  bdciun  16789  bdciin  16790  bdcsuc  16791  bdcriota  16794