Theorem bds 16621

Description: Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 16592; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 16592. (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 16619	. . 3 |- BOUNDED { x | ph }
3		bds.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	3	cbvabv 2359	. . 3 |- { x | ph } = { y | ps }
5	2, 4	bdceqi 16613	. 2 |- BOUNDED { y | ps }
6	5	bdph 16620	1 |- BOUNDED ps

Syntax hints:   -> wi 4   <-> wb 105   {cab 2218  BOUNDED wbd 16582

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-bd0 16583  ax-bdsb 16592

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-bdc 16611

This theorem is referenced by: (None)