Theorem bdcab 16994

Description: A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)

Hypothesis

Ref	Expression
bdcab.1	|- BOUNDED ph

Assertion

Ref	Expression
bdcab	|- BOUNDED { x | ph }

Proof of Theorem bdcab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdcab.1	. . 3 |- BOUNDED ph
2	1	bdab 16983	. 2 |- BOUNDED y e. { x | ph }
3	2	bdelir 16992	1 |- BOUNDED { x | ph }

This proof depends on syntax axioms:   {cab 2224  BOUNDED wbd 16957  BOUNDED wbdc 16985

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1502  ax-bd0 16958  ax-bdsb 16967

This proof depends on definitions:  df-bi 117  df-clab 2225  df-bdc 16986

This theorem is used by:  bds  16996  bdcrab  16997  bdccsb  17005  bdcdif  17006  bdcun  17007  bdcin  17008  bdcpw  17014  bdcsn  17015  bdcuni  17021  bdcint  17022  bdciun  17023  bdciin  17024  bdcriota  17028  bj-bdfindis  17092