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Theorem spime 2408
 Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. Usage of this theorem is discouraged because it depends on ax-13 2391. Check out spimew 1974 for a weaker version requiring less axioms. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (New usage is discouraged.)
Hypotheses
Ref Expression
spime.1 𝑥𝜑
spime.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spime (𝜑 → ∃𝑥𝜓)

Proof of Theorem spime
StepHypRef Expression
1 spime.1 . . . 4 𝑥𝜑
21a1i 11 . . 3 (⊤ → Ⅎ𝑥𝜑)
3 spime.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3spimed 2407 . 2 (⊤ → (𝜑 → ∃𝑥𝜓))
54mptru 1545 1 (𝜑 → ∃𝑥𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ⊤wtru 1539  ∃wex 1781  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2178  ax-13 2391 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-nf 1786 This theorem is referenced by:  spimev  2411  exnel  33121
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