Theorem bdsbc 16884

Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 16885. (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 16848	. 2 |- BOUNDED [ y / x ] ph
3		sbsbc 3055	. 2 |- ( [ y / x ] ph <-> [. y / x ]. ph )
4	2, 3	bd0 16850	1 |- BOUNDED [. y / x ]. ph

This proof depends on syntax axioms:   [wsb 1815   [.wsbc 3051  BOUNDED wbd 16838

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-bd0 16839  ax-bdsb 16848

This proof depends on definitions:  df-bi 117  df-clab 2225  df-cleq 2231  df-clel 2234  df-sbc 3052

This theorem is used by:  bdccsb  16886