Theorem bdsbc 13056

Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 13057. (Contributed by BJ, 16-Oct-2019.)

Hypothesis

Ref	Expression
bdcsbc.1	|- BOUNDED ph

Assertion

Ref	Expression
bdsbc	|- BOUNDED [. y / x ]. ph

Proof of Theorem bdsbc

Step	Hyp	Ref	Expression
1		bdcsbc.1	. . 3 |- BOUNDED ph
2	1	ax-bdsb 13020	. 2 |- BOUNDED [ y / x ] ph
3		sbsbc 2913	. 2 |- ( [ y / x ] ph <-> [. y / x ]. ph )
4	2, 3	bd0 13022	1 |- BOUNDED [. y / x ]. ph

Syntax hints:   [wsb 1735   [.wsbc 2909  BOUNDED wbd 13010

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121  ax-bd0 13011  ax-bdsb 13020

This theorem depends on definitions:  df-bi 116  df-clab 2126  df-cleq 2132  df-clel 2135  df-sbc 2910

This theorem is referenced by:  bdccsb  13058