Theorem bdph 14641

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 14637	. . . 4 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3		df-clab 2164	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
4	2, 3	bd0 14615	. . 3 ⊢ BOUNDED [𝑦 / 𝑥]𝜑
5	4	ax-bdsb 14613	. 2 ⊢ BOUNDED [𝑥 / 𝑦][𝑦 / 𝑥]𝜑
6		sbid2v 1996	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑)
7	5, 6	bd0 14615	1 ⊢ BOUNDED 𝜑

Syntax hints:  [wsb 1762   ∈ wcel 2148  {cab 2163  BOUNDED wbd 14603  BOUNDED wbdc 14631

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-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-bd0 14604  ax-bdsb 14613

This theorem depends on definitions:  df-bi 117  df-sb 1763  df-clab 2164  df-bdc 14632

This theorem is referenced by:  bds  14642