Theorem rabexgf 39183

Description: A version of rabexg 4812 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypothesis

Ref	Expression
rabexgf.1	|- F/_ x A

Assertion

Ref	Expression
rabexgf	|- ( A e. V -> { x e. A | ph } e. _V )

Proof of Theorem rabexgf

Step	Hyp	Ref	Expression
1		df-rab 2921	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		simpl 473	. . . . 5 |- ( ( x e. A /\ ph ) -> x e. A )
3	2	ss2abi 3674	. . . 4 |- { x | ( x e. A /\ ph ) } C_ { x | x e. A }
4		rabexgf.1	. . . . 5 |- F/_ x A
5	4	abid2f 2791	. . . 4 |- { x | x e. A } = A
6	3, 5	sseqtri 3637	. . 3 |- { x | ( x e. A /\ ph ) } C_ A
7	1, 6	eqsstri 3635	. 2 |- { x e. A | ph } C_ A
8		ssexg 4804	. 2 |- ( ( { x e. A | ph } C_ A /\ A e. V ) -> { x e. A | ph } e. _V )
9	7, 8	mpan 706	1 |- ( A e. V -> { x e. A | ph } e. _V )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   {cab 2608   F/_wnfc 2751   {crab 2916   _Vcvv 3200   C_ wss 3574

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:  rabexf  39319  stoweidlem27  40244  stoweidlem35  40252