Theorem bdsbc 16131

Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 16132. (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 16095	. 2 ⊢ BOUNDED [𝑦 / 𝑥]𝜑
3		sbsbc 3012	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	2, 3	bd0 16097	1 ⊢ BOUNDED [𝑦 / 𝑥]𝜑

Syntax hints:  [wsb 1788  [wsbc 3008  BOUNDED wbd 16085

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 1473  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-4 1536  ax-17 1552  ax-ial 1560  ax-ext 2191  ax-bd0 16086  ax-bdsb 16095

This theorem depends on definitions:  df-bi 117  df-clab 2196  df-cleq 2202  df-clel 2205  df-sbc 3009

This theorem is referenced by:  bdccsb  16133