Theorem bdceqir 14367

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 14366) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 14348). (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 2181	. 2 ⊢ 𝐴 = 𝐵
4	1, 3	bdceqi 14366	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1353  BOUNDED wbdc 14363

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159  ax-bd0 14336

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-bdc 14364

This theorem is referenced by:  bdcrab  14375  bdccsb  14383  bdcdif  14384  bdcun  14385  bdcin  14386  bdcnulALT  14389  bdcpw  14392  bdcsn  14393  bdcpr  14394  bdctp  14395  bdcuni  14399  bdcint  14400  bdciun  14401  bdciin  14402  bdcsuc  14403  bdcriota  14406