Theorem bdceq 16163

Description: Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)

Hypothesis

Ref	Expression
bdceq.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
bdceq	⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵)

Proof of Theorem bdceq

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdceq.1	. . . . 5 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2296	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3	2	bdeq 16144	. . 3 ⊢ (BOUNDED 𝑥 ∈ 𝐴 ↔ BOUNDED 𝑥 ∈ 𝐵)
4	3	albii 1516	. 2 ⊢ (∀𝑥BOUNDED 𝑥 ∈ 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐵)
5		df-bdc 16162	. 2 ⊢ (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐴)
6		df-bdc 16162	. 2 ⊢ (BOUNDED 𝐵 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐵)
7	4, 5, 6	3bitr4i 212	1 ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵)

Syntax hints:   ↔ wb 105  ∀wal 1393   = wceq 1395   ∈ wcel 2200  BOUNDED wbd 16133  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:  bdceqi  16164