Theorem sbcex 3055

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 3047	. 2 |- ( [.A / x ].ph <-> A e. {x | ph})
2		elex 2867	. 2 |- (A e. {x | ph} -> A e. _V)
3	1, 2	sylbi 187	1 |- ( [.A / x ].ph -> A e. _V)

Syntax hints:   -> wi 4   e. wcel 1710  {cab 2339  _Vcvv 2859   [.wsbc 3046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861  df-sbc 3047

This theorem is referenced by:  sbcco  3068  sbc5  3070  sbcan  3088  sbcor  3090  sbcal  3093  sbcex2  3095  spesbc  3127  opelopabsb  4697