Theorem bdph 13885

Description: A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)

Hypothesis

Ref	Expression
bdph.1	⊢ BOUNDED {𝑥 ∣ 𝜑}

Assertion

Ref	Expression
bdph	⊢ BOUNDED 𝜑

Proof of Theorem bdph

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdph.1	. . . . 5 ⊢ BOUNDED {𝑥 ∣ 𝜑}
2	1	bdeli 13881	. . . 4 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3		df-clab 2157	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
4	2, 3	bd0 13859	. . 3 ⊢ BOUNDED [𝑦 / 𝑥]𝜑
5	4	ax-bdsb 13857	. 2 ⊢ BOUNDED [𝑥 / 𝑦][𝑦 / 𝑥]𝜑
6		sbid2v 1989	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑)
7	5, 6	bd0 13859	1 ⊢ BOUNDED 𝜑

Syntax hints:  [wsb 1755   ∈ wcel 2141  {cab 2156  BOUNDED wbd 13847  BOUNDED wbdc 13875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-bd0 13848  ax-bdsb 13857

This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-bdc 13876

This theorem is referenced by:  bds  13886