Theorem sbcex 2886

Description: By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
sbcex	|- ( [. A / x ]. ph -> A e. _V )

Proof of Theorem sbcex

Step	Hyp	Ref	Expression
1		df-sbc 2879	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
2		elex 2668	. 2 |- ( A e. { x | ph } -> A e. _V )
3	1, 2	sylbi 120	1 |- ( [. A / x ]. ph -> A e. _V )

Syntax hints:   -> wi 4   e. wcel 1463   {cab 2101   _Vcvv 2657   [.wsbc 2878

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-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2659  df-sbc 2879

This theorem is referenced by:  sbcco  2899  sbc5  2901  sbcan  2919  sbcor  2921  sbcn1  2924  sbcim1  2925  sbcbi1  2926  sbcal  2928  sbcex2  2930  sbcel1v  2939  sbcel21v  2941  sbcimdv  2942  sbcrext  2954  spesbc  2962  csbprc  3374  opelopabsb  4142