Theorem bdceq 10633

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 2145	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3	2	bdeq 10614	. . 3 ⊢ (BOUNDED 𝑥 ∈ 𝐴 ↔ BOUNDED 𝑥 ∈ 𝐵)
4	3	albii 1399	. 2 ⊢ (∀𝑥BOUNDED 𝑥 ∈ 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐵)
5		df-bdc 10632	. 2 ⊢ (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐴)
6		df-bdc 10632	. 2 ⊢ (BOUNDED 𝐵 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐵)
7	4, 5, 6	3bitr4i 210	1 ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵)

Syntax hints:   ↔ wb 103  ∀wal 1282   = wceq 1284   ∈ wcel 1433  BOUNDED wbd 10603  BOUNDED wbdc 10631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063  ax-bd0 10604

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077  df-bdc 10632

This theorem is referenced by:  bdceqi  10634