Theorem bdceqi 14598

Description: A class equal to a bounded one is bounded. Note the use of ax-ext 2159. See also bdceqir 14599. (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 14597	. 2 ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵)
4	1, 3	mpbi 145	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1353  BOUNDED wbdc 14595

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159  ax-bd0 14568

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-bdc 14596

This theorem is referenced by:  bdceqir  14599  bds  14606  bdcuni  14631