Theorem bdph 15860

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 15856	. . . 4 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3		df-clab 2193	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
4	2, 3	bd0 15834	. . 3 ⊢ BOUNDED [𝑦 / 𝑥]𝜑
5	4	ax-bdsb 15832	. 2 ⊢ BOUNDED [𝑥 / 𝑦][𝑦 / 𝑥]𝜑
6		sbid2v 2025	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑)
7	5, 6	bd0 15834	1 ⊢ BOUNDED 𝜑

Syntax hints:  [wsb 1786   ∈ wcel 2177  {cab 2192  BOUNDED wbd 15822  BOUNDED wbdc 15850

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-bd0 15823  ax-bdsb 15832

This theorem depends on definitions:  df-bi 117  df-sb 1787  df-clab 2193  df-bdc 15851

This theorem is referenced by:  bds  15861