Theorem bdceqir 16614

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16613) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 16595). (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 2236	. 2 ⊢ 𝐴 = 𝐵
4	1, 3	bdceqi 16613	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1398  BOUNDED wbdc 16610

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2214  ax-bd0 16583

This theorem depends on definitions:  df-bi 117  df-cleq 2225  df-clel 2228  df-bdc 16611

This theorem is referenced by:  bdcrab  16622  bdccsb  16630  bdcdif  16631  bdcun  16632  bdcin  16633  bdcnulALT  16636  bdcpw  16639  bdcsn  16640  bdcpr  16641  bdctp  16642  bdcuni  16646  bdcint  16647  bdciun  16648  bdciin  16649  bdcsuc  16650  bdcriota  16653