Theorem intexabim 4154

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 2165	. . 3 |- ( x e. { x | ph } <-> ph )
2	1	exbii 1605	. 2 |- ( E. x x e. { x | ph } <-> E. x ph )
3		nfsab1 2167	. . . 4 |- F/ x y e. { x | ph }
4		nfv 1528	. . . 4 |- F/ y x e. { x | ph }
5		eleq1 2240	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
6	3, 4, 5	cbvex 1756	. . 3 |- ( E. y y e. { x | ph } <-> E. x x e. { x | ph } )
7		inteximm 4151	. . 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 1492   e. wcel 2148   {cab 2163   _Vcvv 2739   |^|cint 3846

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4123

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-ss 3144  df-int 3847

This theorem is referenced by:  intexrabim  4155  omex  4594