Theorem bdceqir 15817

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 15816) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 15798). (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 15816	1 |- BOUNDED B

Syntax hints:   = wceq 1373  BOUNDED wbdc 15813

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 15786

This theorem depends on definitions:  df-bi 117  df-cleq 2198  df-clel 2201  df-bdc 15814

This theorem is referenced by:  bdcrab  15825  bdccsb  15833  bdcdif  15834  bdcun  15835  bdcin  15836  bdcnulALT  15839  bdcpw  15842  bdcsn  15843  bdcpr  15844  bdctp  15845  bdcuni  15849  bdcint  15850  bdciun  15851  bdciin  15852  bdcsuc  15853  bdcriota  15856