New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  sbcth2 GIF version

Theorem sbcth2 3129
 Description: A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Hypothesis
Ref Expression
sbcth2.1 (x Bφ)
Assertion
Ref Expression
sbcth2 (A B → [̣A / xφ)
Distinct variable group:   x,B
Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem sbcth2
StepHypRef Expression
1 sbcth2.1 . . 3 (x Bφ)
21rgen 2679 . 2 x B φ
3 rspsbc 3124 . 2 (A B → (x B φ → [̣A / xφ))
42, 3mpi 16 1 (A B → [̣A / xφ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2614  [̣wsbc 3046 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-sbc 3047 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator