Theorem bdceqir 16262

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16261) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 16243). (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 2233	. 2 ⊢ 𝐴 = 𝐵
4	1, 3	bdceqi 16261	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1395  BOUNDED wbdc 16258

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211  ax-bd0 16231

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-bdc 16259

This theorem is referenced by:  bdcrab  16270  bdccsb  16278  bdcdif  16279  bdcun  16280  bdcin  16281  bdcnulALT  16284  bdcpw  16287  bdcsn  16288  bdcpr  16289  bdctp  16290  bdcuni  16294  bdcint  16295  bdciun  16296  bdciin  16297  bdcsuc  16298  bdcriota  16301