Theorem bdsbc 15007

Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 15008. (Contributed by BJ, 16-Oct-2019.)

Hypothesis

Ref	Expression
bdcsbc.1	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bdsbc	⊢ BOUNDED [𝑦 / 𝑥]𝜑

Proof of Theorem bdsbc

Step	Hyp	Ref	Expression
1		bdcsbc.1	. . 3 ⊢ BOUNDED 𝜑
2	1	ax-bdsb 14971	. 2 ⊢ BOUNDED [𝑦 / 𝑥]𝜑
3		sbsbc 2981	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	2, 3	bd0 14973	1 ⊢ BOUNDED [𝑦 / 𝑥]𝜑

Syntax hints:  [wsb 1773  [wsbc 2977  BOUNDED wbd 14961

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171  ax-bd0 14962  ax-bdsb 14971

This theorem depends on definitions:  df-bi 117  df-clab 2176  df-cleq 2182  df-clel 2185  df-sbc 2978

This theorem is referenced by:  bdccsb  15009