Theorem bdceqir 16375

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16374) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 16356). (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 16374	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1395  BOUNDED wbdc 16371

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 16344

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-bdc 16372

This theorem is referenced by:  bdcrab  16383  bdccsb  16391  bdcdif  16392  bdcun  16393  bdcin  16394  bdcnulALT  16397  bdcpw  16400  bdcsn  16401  bdcpr  16402  bdctp  16403  bdcuni  16407  bdcint  16408  bdciun  16409  bdciin  16410  bdcsuc  16411  bdcriota  16414