Theorem bdceqir 13726

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 13725) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 13707). (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 2169	. 2 ⊢ 𝐴 = 𝐵
4	1, 3	bdceqi 13725	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1343  BOUNDED wbdc 13722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147  ax-bd0 13695

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-clel 2161  df-bdc 13723

This theorem is referenced by:  bdcrab  13734  bdccsb  13742  bdcdif  13743  bdcun  13744  bdcin  13745  bdcnulALT  13748  bdcpw  13751  bdcsn  13752  bdcpr  13753  bdctp  13754  bdcuni  13758  bdcint  13759  bdciun  13760  bdciin  13761  bdcsuc  13762  bdcriota  13765