Theorem rab2ex 4278

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 4277	. 2 |- B e. _V
4	3	rabex 4275	1 |- { x e. B | ph } e. _V

Syntax hints:   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4244

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-in 3226  df-ss 3233

This theorem is referenced by: (None)