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Theorem spimew 1979
Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 22-Oct-2023.)
Hypotheses
Ref Expression
spimew.1 (𝜑 → ∀𝑥𝜑)
spimew.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spimew (𝜑 → ∃𝑥𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem spimew
StepHypRef Expression
1 ax6v 1976 . 2 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2 spimew.1 . 2 (𝜑 → ∀𝑥𝜑)
3 spimew.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
43speimfw 1971 . 2 (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))
51, 2, 4mpsyl 68 1 (𝜑 → ∃𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1540  wex 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-6 1975
This theorem depends on definitions:  df-bi 210  df-ex 1787
This theorem is referenced by:  speiv  1981  spimevw  2006  bj-cbvexiw  34507
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