Theorem bdceqir 15576

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 15575) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 15557). (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 15575	1 |- BOUNDED B

Syntax hints:   = wceq 1364  BOUNDED wbdc 15572

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 15545

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192  df-bdc 15573

This theorem is referenced by:  bdcrab  15584  bdccsb  15592  bdcdif  15593  bdcun  15594  bdcin  15595  bdcnulALT  15598  bdcpw  15601  bdcsn  15602  bdcpr  15603  bdctp  15604  bdcuni  15608  bdcint  15609  bdciun  15610  bdciin  15611  bdcsuc  15612  bdcriota  15615