Theorem bdceqi 16164

Description: A class equal to a bounded one is bounded. Note the use of ax-ext 2211. See also bdceqir 16165. (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 16163	. 2 ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵)
4	1, 3	mpbi 145	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1395  BOUNDED wbdc 16161

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211  ax-bd0 16134

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-bdc 16162

This theorem is referenced by:  bdceqir  16165  bds  16172  bdcuni  16197