Theorem bdph 10826

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 10822	. . . 4 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3		df-clab 2069	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
4	2, 3	bd0 10800	. . 3 ⊢ BOUNDED [𝑦 / 𝑥]𝜑
5	4	ax-bdsb 10798	. 2 ⊢ BOUNDED [𝑥 / 𝑦][𝑦 / 𝑥]𝜑
6		sbid2v 1914	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑)
7	5, 6	bd0 10800	1 ⊢ BOUNDED 𝜑

Syntax hints:   ∈ wcel 1434  [wsb 1686  {cab 2068  BOUNDED wbd 10788  BOUNDED wbdc 10816

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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-bd0 10789  ax-bdsb 10798

This theorem depends on definitions:  df-bi 115  df-sb 1687  df-clab 2069  df-bdc 10817

This theorem is referenced by:  bds  10827