Theorem bdsbc 13444

Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 13445. (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 13408	. 2 ⊢ BOUNDED [𝑦 / 𝑥]𝜑
3		sbsbc 2941	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	2, 3	bd0 13410	1 ⊢ BOUNDED [𝑦 / 𝑥]𝜑

Syntax hints:  [wsb 1742  [wsbc 2937  BOUNDED wbd 13398

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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-ext 2139  ax-bd0 13399  ax-bdsb 13408

This theorem depends on definitions:  df-bi 116  df-clab 2144  df-cleq 2150  df-clel 2153  df-sbc 2938

This theorem is referenced by:  bdccsb  13446