Theorem bdsbcALT 15515

Description: Alternate proof of bdsbc 15514. (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 15494	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3		df-sbc 2990	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑})
4	2, 3	bd0r 15481	1 ⊢ BOUNDED [𝑦 / 𝑥]𝜑

Syntax hints:   ∈ wcel 2167  {cab 2182  [wsbc 2989  BOUNDED wbd 15468

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-bd0 15469  ax-bdsb 15478

This theorem depends on definitions:  df-bi 117  df-clab 2183  df-sbc 2990

This theorem is referenced by: (None)