Theorem bdsbc 11237

Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 11238. (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 11201	. 2 |- BOUNDED [ y / x ] ph
3		sbsbc 2833	. 2 |- ( [ y / x ] ph <-> [. y / x ]. ph )
4	2, 3	bd0 11203	1 |- BOUNDED [. y / x ]. ph

Syntax hints:   [wsb 1689   [.wsbc 2829  BOUNDED wbd 11191

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067  ax-bd0 11192  ax-bdsb 11201

This theorem depends on definitions:  df-bi 115  df-clab 2072  df-cleq 2078  df-clel 2081  df-sbc 2830

This theorem is referenced by:  bdccsb  11239