Theorem bdsbcALT 10650

Description: Alternate proof of bdsbc 10649. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bdcsbc.1	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bdsbcALT	⊢ BOUNDED [𝑦 / 𝑥]𝜑

Proof of Theorem bdsbcALT

Step	Hyp	Ref	Expression
1		bdcsbc.1	. . 3 ⊢ BOUNDED 𝜑
2	1	bdab 10629	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3		df-sbc 2816	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑})
4	2, 3	bd0r 10616	1 ⊢ BOUNDED [𝑦 / 𝑥]𝜑

Syntax hints:   ∈ wcel 1433  {cab 2067  [wsbc 2815  BOUNDED wbd 10603

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-bd0 10604  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-clab 2068  df-sbc 2816

This theorem is referenced by: (None)