Theorem sbcex 2995

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 2987	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
2		elex 2771	. 2 |- ( A e. { x | ph } -> A e. _V )
3	1, 2	sylbi 121	1 |- ( [. A / x ]. ph -> A e. _V )

Syntax hints:   -> wi 4   e. wcel 2164   {cab 2179   _Vcvv 2760   [.wsbc 2986

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-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-v 2762  df-sbc 2987

This theorem is referenced by:  sbcco  3008  sbc5  3010  sbcan  3029  sbcor  3031  sbcn1  3034  sbcim1  3035  sbcbi1  3036  sbcal  3038  sbcex2  3040  sbcel1v  3049  sbcel21v  3051  sbcimdv  3052  sbcrext  3064  spesbc  3072  csbprc  3493  opelopabsb  4291