Theorem bdceqi 10634

Description: A class equal to a bounded one is bounded. Note the use of ax-ext 2063. See also bdceqir 10635. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bdceqi.min	⊢ BOUNDED 𝐴
bdceqi.maj	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
bdceqi	⊢ BOUNDED 𝐵

Proof of Theorem bdceqi

Step	Hyp	Ref	Expression
1		bdceqi.min	. 2 ⊢ BOUNDED 𝐴
2		bdceqi.maj	. . 3 ⊢ 𝐴 = 𝐵
3	2	bdceq 10633	. 2 ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵)
4	1, 3	mpbi 143	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1284  BOUNDED wbdc 10631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063  ax-bd0 10604

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077  df-bdc 10632

This theorem is referenced by:  bdceqir  10635  bds  10642  bdcuni  10667