Theorem bdceqir 15406

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 15405) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 15387). (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 2197	. 2 ⊢ 𝐴 = 𝐵
4	1, 3	bdceqi 15405	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1364  BOUNDED wbdc 15402

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 2175  ax-bd0 15375

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189  df-bdc 15403

This theorem is referenced by:  bdcrab  15414  bdccsb  15422  bdcdif  15423  bdcun  15424  bdcin  15425  bdcnulALT  15428  bdcpw  15431  bdcsn  15432  bdcpr  15433  bdctp  15434  bdcuni  15438  bdcint  15439  bdciun  15440  bdciin  15441  bdcsuc  15442  bdcriota  15445