Theorem bdph 13219

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 13215	. . . 4 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3		df-clab 2127	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
4	2, 3	bd0 13193	. . 3 ⊢ BOUNDED [𝑦 / 𝑥]𝜑
5	4	ax-bdsb 13191	. 2 ⊢ BOUNDED [𝑥 / 𝑦][𝑦 / 𝑥]𝜑
6		sbid2v 1972	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑)
7	5, 6	bd0 13193	1 ⊢ BOUNDED 𝜑

Syntax hints:   ∈ wcel 1481  [wsb 1736  {cab 2126  BOUNDED wbd 13181  BOUNDED wbdc 13209

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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-bd0 13182  ax-bdsb 13191

This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-bdc 13210

This theorem is referenced by:  bds  13220