Theorem intexabim 4185

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 2184	. . 3 |- ( x e. { x | ph } <-> ph )
2	1	exbii 1619	. 2 |- ( E. x x e. { x | ph } <-> E. x ph )
3		nfsab1 2186	. . . 4 |- F/ x y e. { x | ph }
4		nfv 1542	. . . 4 |- F/ y x e. { x | ph }
5		eleq1 2259	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
6	3, 4, 5	cbvex 1770	. . 3 |- ( E. y y e. { x | ph } <-> E. x x e. { x | ph } )
7		inteximm 4182	. . 3 |- ( E. y y e. { x | ph } -> |^| { x | ph } e. _V )
8	6, 7	sylbir 135	. 2 |- ( E. x x e. { x | ph } -> |^| { x | ph } e. _V )
9	2, 8	sylbir 135	1 |- ( E. x ph -> |^| { x | ph } e. _V )

Syntax hints:   -> wi 4   E.wex 1506   e. wcel 2167   {cab 2182   _Vcvv 2763   |^|cint 3874

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-sep 4151

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-int 3875

This theorem is referenced by:  intexrabim  4186  omex  4629