Theorem rab2ex 4176

Description: A class abstraction based on a class abstraction based on a set is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)

Hypotheses

Ref	Expression
rab2ex.1	|- B = { y e. A | ps }
rab2ex.2	|- A e. _V

Assertion

Ref	Expression
rab2ex	|- { x e. B | ph } e. _V

Distinct variable groups:   x, B   y, A

Allowed substitution hints:   ph( x, y)   ps( x, y)   A( x)   B( y)

Proof of Theorem rab2ex

Step	Hyp	Ref	Expression
1		rab2ex.1	. . 3 |- B = { y e. A | ps }
2		rab2ex.2	. . 3 |- A e. _V
3	1, 2	rabex2 4175	. 2 |- B e. _V
4	3	rabex 4173	1 |- { x e. B | ph } e. _V

Syntax hints:   = wceq 1364   e. wcel 2164   {crab 2476   _Vcvv 2760

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-sep 4147

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-in 3159  df-ss 3166

This theorem is referenced by: (None)