Theorem bdceqi 15573

Description: A class equal to a bounded one is bounded. Note the use of ax-ext 2178. See also bdceqir 15574. (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 15572	. 2 ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵)
4	1, 3	mpbi 145	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1364  BOUNDED wbdc 15570

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-bd0 15543

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192  df-bdc 15571

This theorem is referenced by:  bdceqir  15574  bds  15581  bdcuni  15606