Theorem bdsbc 13893

Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 13894. (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 13857	. 2 ⊢ BOUNDED [𝑦 / 𝑥]𝜑
3		sbsbc 2959	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	2, 3	bd0 13859	1 ⊢ BOUNDED [𝑦 / 𝑥]𝜑

Syntax hints:  [wsb 1755  [wsbc 2955  BOUNDED wbd 13847

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152  ax-bd0 13848  ax-bdsb 13857

This theorem depends on definitions:  df-bi 116  df-clab 2157  df-cleq 2163  df-clel 2166  df-sbc 2956

This theorem is referenced by:  bdccsb  13895