Theorem bds 14688

Description: Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 14659; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 14659. (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 14686	. . 3 |- BOUNDED { x | ph }
3		bds.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	3	cbvabv 2302	. . 3 |- { x | ph } = { y | ps }
5	2, 4	bdceqi 14680	. 2 |- BOUNDED { y | ps }
6	5	bdph 14687	1 |- BOUNDED ps

Syntax hints:   -> wi 4   <-> wb 105   {cab 2163  BOUNDED wbd 14649

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14650  ax-bdsb 14659

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-bdc 14678

This theorem is referenced by: (None)