Theorem bdceqir 14599

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 14598) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 14580). (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 14598	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1353  BOUNDED wbdc 14595

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 14568

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-bdc 14596

This theorem is referenced by:  bdcrab  14607  bdccsb  14615  bdcdif  14616  bdcun  14617  bdcin  14618  bdcnulALT  14621  bdcpw  14624  bdcsn  14625  bdcpr  14626  bdctp  14627  bdcuni  14631  bdcint  14632  bdciun  14633  bdciin  14634  bdcsuc  14635  bdcriota  14638