Theorem bdceqir 16560

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16559) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 16541). (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 2235	. 2 |- A = B
4	1, 3	bdceqi 16559	1 |- BOUNDED B

Syntax hints:   = wceq 1398  BOUNDED wbdc 16556

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 2213  ax-bd0 16529

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227  df-bdc 16557

This theorem is referenced by:  bdcrab  16568  bdccsb  16576  bdcdif  16577  bdcun  16578  bdcin  16579  bdcnulALT  16582  bdcpw  16585  bdcsn  16586  bdcpr  16587  bdctp  16588  bdcuni  16592  bdcint  16593  bdciun  16594  bdciin  16595  bdcsuc  16596  bdcriota  16599