Theorem bdceqir 15742

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 15741) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 15723). (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 2208	. 2 ⊢ 𝐴 = 𝐵
4	1, 3	bdceqi 15741	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1372  BOUNDED wbdc 15738

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186  ax-bd0 15711

This theorem depends on definitions:  df-bi 117  df-cleq 2197  df-clel 2200  df-bdc 15739

This theorem is referenced by:  bdcrab  15750  bdccsb  15758  bdcdif  15759  bdcun  15760  bdcin  15761  bdcnulALT  15764  bdcpw  15767  bdcsn  15768  bdcpr  15769  bdctp  15770  bdcuni  15774  bdcint  15775  bdciun  15776  bdciin  15777  bdcsuc  15778  bdcriota  15781