Theorem intexabim 4131

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 2153	. . 3 |- ( x e. { x | ph } <-> ph )
2	1	exbii 1593	. 2 |- ( E. x x e. { x | ph } <-> E. x ph )
3		nfsab1 2155	. . . 4 |- F/ x y e. { x | ph }
4		nfv 1516	. . . 4 |- F/ y x e. { x | ph }
5		eleq1 2229	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
6	3, 4, 5	cbvex 1744	. . 3 |- ( E. y y e. { x | ph } <-> E. x x e. { x | ph } )
7		inteximm 4128	. . 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 1480   e. wcel 2136   {cab 2151   _Vcvv 2726   |^|cint 3824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-sep 4100

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-int 3825

This theorem is referenced by:  intexrabim  4132  omex  4570