Theorem intexabim 4009

Description: The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)

Assertion

Ref	Expression
intexabim	|- ( E. x ph -> |^| { x | ph } e. _V )

Proof of Theorem intexabim

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		abid 2083	. . 3 |- ( x e. { x | ph } <-> ph )
2	1	exbii 1548	. 2 |- ( E. x x e. { x | ph } <-> E. x ph )
3		nfsab1 2085	. . . 4 |- F/ x y e. { x | ph }
4		nfv 1473	. . . 4 |- F/ y x e. { x | ph }
5		eleq1 2157	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
6	3, 4, 5	cbvex 1693	. . 3 |- ( E. y y e. { x | ph } <-> E. x x e. { x | ph } )
7		inteximm 4006	. . 3 |- ( E. y y e. { x | ph } -> |^| { x | ph } e. _V )
8	6, 7	sylbir 134	. 2 |- ( E. x x e. { x | ph } -> |^| { x | ph } e. _V )
9	2, 8	sylbir 134	1 |- ( E. x ph -> |^| { x | ph } e. _V )

Syntax hints:   -> wi 4   E.wex 1433   e. wcel 1445   {cab 2081   _Vcvv 2633   |^|cint 3710

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026  df-int 3711

This theorem is referenced by:  intexrabim  4010  omex  4436