Theorem bdceqir 14532

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 14531) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 14513). (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 14531	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1353  BOUNDED wbdc 14528

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 14501

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-bdc 14529

This theorem is referenced by:  bdcrab  14540  bdccsb  14548  bdcdif  14549  bdcun  14550  bdcin  14551  bdcnulALT  14554  bdcpw  14557  bdcsn  14558  bdcpr  14559  bdctp  14560  bdcuni  14564  bdcint  14565  bdciun  14566  bdciin  14567  bdcsuc  14568  bdcriota  14571