Theorem bdsbc 15993

Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 15994. (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 15957	. 2 |- BOUNDED [ y / x ] ph
3		sbsbc 3009	. 2 |- ( [ y / x ] ph <-> [. y / x ]. ph )
4	2, 3	bd0 15959	1 |- BOUNDED [. y / x ]. ph

Syntax hints:   [wsb 1786   [.wsbc 3005  BOUNDED wbd 15947

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-ext 2189  ax-bd0 15948  ax-bdsb 15957

This theorem depends on definitions:  df-bi 117  df-clab 2194  df-cleq 2200  df-clel 2203  df-sbc 3006

This theorem is referenced by:  bdccsb  15995