Theorem bdceqir 14993

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 14992) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 14974). (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 2193	. 2 |- A = B
4	1, 3	bdceqi 14992	1 |- BOUNDED B

Syntax hints:   = wceq 1364  BOUNDED wbdc 14989

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171  ax-bd0 14962

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185  df-bdc 14990

This theorem is referenced by:  bdcrab  15001  bdccsb  15009  bdcdif  15010  bdcun  15011  bdcin  15012  bdcnulALT  15015  bdcpw  15018  bdcsn  15019  bdcpr  15020  bdctp  15021  bdcuni  15025  bdcint  15026  bdciun  15027  bdciin  15028  bdcsuc  15029  bdcriota  15032