Theorem bds 11211

Description: Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 11182; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 11182. (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 11209	. . 3 |- BOUNDED { x | ph }
3		bds.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	3	cbvabv 2208	. . 3 |- { x | ph } = { y | ps }
5	2, 4	bdceqi 11203	. 2 |- BOUNDED { y | ps }
6	5	bdph 11210	1 |- BOUNDED ps

Syntax hints:   -> wi 4   <-> wb 103   {cab 2071  BOUNDED wbd 11172

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-bd0 11173  ax-bdsb 11182

This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-bdc 11201

This theorem is referenced by: (None)