Theorem sbcex 3054

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 3046	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
2		elex 2827	. 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 2205   {cab 2220   _Vcvv 2815   [.wsbc 3045

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-v 2817  df-sbc 3046

This theorem is referenced by:  sbcco  3067  sbc5  3069  sbcan  3088  sbcor  3090  sbcn1  3093  sbcim1  3094  sbcbi1  3095  sbcal  3097  sbcex2  3099  sbcel1v  3108  sbcel21v  3110  sbcimdv  3111  sbcrext  3123  spesbc  3132  csbprc  3558  opelopabsb  4383