Theorem bdceqi 13030

Description: A class equal to a bounded one is bounded. Note the use of ax-ext 2119. See also bdceqir 13031. (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 13029	. 2 ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵)
4	1, 3	mpbi 144	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1331  BOUNDED wbdc 13027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119  ax-bd0 13000

This theorem depends on definitions:  df-bi 116  df-cleq 2130  df-clel 2133  df-bdc 13028

This theorem is referenced by:  bdceqir  13031  bds  13038  bdcuni  13063