Theorem intexabim 3994

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 2077	. . 3 |- ( x e. { x | ph } <-> ph )
2	1	exbii 1542	. 2 |- ( E. x x e. { x | ph } <-> E. x ph )
3		nfsab1 2079	. . . 4 |- F/ x y e. { x | ph }
4		nfv 1467	. . . 4 |- F/ y x e. { x | ph }
5		eleq1 2151	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
6	3, 4, 5	cbvex 1687	. . 3 |- ( E. y y e. { x | ph } <-> E. x x e. { x | ph } )
7		inteximm 3991	. . 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 1427   e. wcel 1439   {cab 2075   _Vcvv 2620   |^|cint 3694

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 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006  df-ss 3013  df-int 3695

This theorem is referenced by:  intexrabim  3995  omex  4421