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

Theorem spimw 1668
 Description: Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
Hypotheses
Ref Expression
spimw.1 ψx ¬ ψ)
spimw.2 (x = y → (φψ))
Assertion
Ref Expression
spimw (xφψ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem spimw
StepHypRef Expression
1 ax9v 1655 . 2 ¬ x ¬ x = y
2 spimw.1 . . 3 ψx ¬ ψ)
3 spimw.2 . . 3 (x = y → (φψ))
42, 3spimfw 1646 . 2 x ¬ x = y → (xφψ))
51, 4ax-mp 5 1 (xφψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540 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-9 1654 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by:  spimvw  1669  spnfw  1670  cbvaliw  1673  spfw  1691
 Copyright terms: Public domain W3C validator