Theorem bdsbc 13740

Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 13741. (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 13704	. 2 |- BOUNDED [ y / x ] ph
3		sbsbc 2955	. 2 |- ( [ y / x ] ph <-> [. y / x ]. ph )
4	2, 3	bd0 13706	1 |- BOUNDED [. y / x ]. ph

Syntax hints:   [wsb 1750   [.wsbc 2951  BOUNDED wbd 13694

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147  ax-bd0 13695  ax-bdsb 13704

This theorem depends on definitions:  df-bi 116  df-clab 2152  df-cleq 2158  df-clel 2161  df-sbc 2952

This theorem is referenced by:  bdccsb  13742