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Theorem spime 2380
Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. See spimew 1967 and spimevw 1990 for weaker versions requiring fewer axioms. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) Usage of this theorem is discouraged because it depends on ax-13 2363. Use spimefv 2183 instead. (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 2379 . 2 (⊤ → (𝜑 → ∃𝑥𝜓))
54mptru 1540 1 (𝜑 → ∃𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wtru 1534  wex 1773  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163  ax-13 2363
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-nf 1778
This theorem is referenced by:  spimev  2383  exnel  35270
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