Theorem bds 13693

Description: Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 13664; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 13664. (Contributed by BJ, 19-Nov-2019.)

Hypotheses

Ref	Expression
bds.bd	|- BOUNDED ph
bds.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
bds	|- BOUNDED ps

Distinct variable groups:   ps, x   ph, y

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem bds

Step	Hyp	Ref	Expression
1		bds.bd	. . . 4 |- BOUNDED ph
2	1	bdcab 13691	. . 3 |- BOUNDED { x | ph }
3		bds.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	3	cbvabv 2290	. . 3 |- { x | ph } = { y | ps }
5	2, 4	bdceqi 13685	. 2 |- BOUNDED { y | ps }
6	5	bdph 13692	1 |- BOUNDED ps

Syntax hints:   -> wi 4   <-> wb 104   {cab 2151  BOUNDED wbd 13654

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-bd0 13655  ax-bdsb 13664

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-bdc 13683

This theorem is referenced by: (None)