Theorem bdceqir 15713

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 15712) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 15694). (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 2208	. 2 ⊢ 𝐴 = 𝐵
4	1, 3	bdceqi 15712	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1372  BOUNDED wbdc 15709

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186  ax-bd0 15682

This theorem depends on definitions:  df-bi 117  df-cleq 2197  df-clel 2200  df-bdc 15710

This theorem is referenced by:  bdcrab  15721  bdccsb  15729  bdcdif  15730  bdcun  15731  bdcin  15732  bdcnulALT  15735  bdcpw  15738  bdcsn  15739  bdcpr  15740  bdctp  15741  bdcuni  15745  bdcint  15746  bdciun  15747  bdciin  15748  bdcsuc  15749  bdcriota  15752