Theorem rab2ex 4816

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 4815	. 2 |- B e. _V
4	3	rabex 4813	1 |- { x e. B | ph } e. _V

Syntax hints:   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588

This theorem is referenced by:  gsumbagdiag  19376  psrlidm  19403  psrridm  19404  psrass1  19405  mdegmullem  23838  vtxdginducedm1  26439