Theorem rab2ex 4199

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 4198	. 2 |- B e. _V
4	3	rabex 4196	1 |- { x e. B | ph } e. _V

Syntax hints:   = wceq 1373   e. wcel 2177   {crab 2489   _Vcvv 2773

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-sep 4170

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rab 2494  df-v 2775  df-in 3176  df-ss 3183

This theorem is referenced by: (None)