New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  sb8 GIF version

Theorem sb8 2092
 Description: Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sb5rf.1 yφ
Assertion
Ref Expression
sb8 (xφy[y / x]φ)

Proof of Theorem sb8
StepHypRef Expression
1 sb5rf.1 . . . 4 yφ
21nfal 1842 . . 3 yxφ
3 stdpc4 2024 . . 3 (xφ → [y / x]φ)
42, 3alrimi 1765 . 2 (xφy[y / x]φ)
51nfs1 2044 . . 3 x[y / x]φ
6 stdpc7 1917 . . 3 (y = x → ([y / x]φφ))
75, 1, 6cbv3 1982 . 2 (y[y / x]φxφ)
84, 7impbii 180 1 (xφy[y / x]φ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544  [wsb 1648 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  sb8e  2093  sbhb  2107  sbnf2  2108  sb8eu  2222  sb8iota  4346
 Copyright terms: Public domain W3C validator