Theorem sbcex 3056

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]̣φ → A ∈ V)

Proof of Theorem sbcex

Step	Hyp	Ref	Expression
1		df-sbc 3048	. 2 ⊢ ([̣A / x]̣φ ↔ A ∈ {x ∣ φ})
2		elex 2868	. 2 ⊢ (A ∈ {x ∣ φ} → A ∈ V)
3	1, 2	sylbi 187	1 ⊢ ([̣A / x]̣φ → A ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1710  {cab 2339  Vcvv 2860  [̣wsbc 3047

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 2862  df-sbc 3048

This theorem is referenced by:  sbcco  3069  sbc5  3071  sbcan  3089  sbcor  3091  sbcal  3094  sbcex2  3096  spesbc  3128  opelopabsb  4698