Theorem bdceqir 15979

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 15978) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 15960). (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 2211	. 2 ⊢ 𝐴 = 𝐵
4	1, 3	bdceqi 15978	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1373  BOUNDED wbdc 15975

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-ext 2189  ax-bd0 15948

This theorem depends on definitions:  df-bi 117  df-cleq 2200  df-clel 2203  df-bdc 15976

This theorem is referenced by:  bdcrab  15987  bdccsb  15995  bdcdif  15996  bdcun  15997  bdcin  15998  bdcnulALT  16001  bdcpw  16004  bdcsn  16005  bdcpr  16006  bdctp  16007  bdcuni  16011  bdcint  16012  bdciun  16013  bdciin  16014  bdcsuc  16015  bdcriota  16018