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

Theorem speimfw 1645
 Description: Specialization, with additional weakening to allow bundling of x and y. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.)
Hypothesis
Ref Expression
speimfw.2 (x = y → (φψ))
Assertion
Ref Expression
speimfw x ¬ x = y → (xφxψ))

Proof of Theorem speimfw
StepHypRef Expression
1 speimfw.2 . . 3 (x = y → (φψ))
21eximi 1576 . 2 (x x = yx(φψ))
3 df-ex 1542 . 2 (x x = y ↔ ¬ x ¬ x = y)
4 19.35 1600 . 2 (x(φψ) ↔ (xφxψ))
52, 3, 43imtr3i 256 1 x ¬ x = y → (xφxψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by:  spimfw  1646  19.2OLD  1700
 Copyright terms: Public domain W3C validator