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Theorem spime 1976
 Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.)
Hypotheses
Ref Expression
spime.1 xφ
spime.2 (x = y → (φψ))
Assertion
Ref Expression
spime (φxψ)

Proof of Theorem spime
StepHypRef Expression
1 spime.1 . . . . 5 xφ
21nfn 1793 . . . 4 x ¬ φ
3 spime.2 . . . . 5 (x = y → (φψ))
43con3d 125 . . . 4 (x = y → (¬ ψ → ¬ φ))
52, 4spim 1975 . . 3 (x ¬ ψ → ¬ φ)
65con2i 112 . 2 (φ → ¬ x ¬ ψ)
7 df-ex 1542 . 2 (xψ ↔ ¬ x ¬ ψ)
86, 7sylibr 203 1 (φxψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544 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 This theorem is referenced by:  spimed  1977  spimev  1999
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