Theorem bdceqir 15598

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 15597) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 15579). (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 2200	. 2 ⊢ 𝐴 = 𝐵
4	1, 3	bdceqi 15597	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1364  BOUNDED wbdc 15594

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-bd0 15567

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192  df-bdc 15595

This theorem is referenced by:  bdcrab  15606  bdccsb  15614  bdcdif  15615  bdcun  15616  bdcin  15617  bdcnulALT  15620  bdcpw  15623  bdcsn  15624  bdcpr  15625  bdctp  15626  bdcuni  15630  bdcint  15631  bdciun  15632  bdciin  15633  bdcsuc  15634  bdcriota  15637