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Theorem spimeh 2082
Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
Hypotheses
Ref Expression
spimeh.1 (𝜑 → ∀𝑥𝜑)
spimeh.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spimeh (𝜑 → ∃𝑥𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem spimeh
StepHypRef Expression
1 spimeh.1 . 2 (𝜑 → ∀𝑥𝜑)
2 ax6ev 2058 . . . 4 𝑥 𝑥 = 𝑦
3 spimeh.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3eximii 1911 . . 3 𝑥(𝜑𝜓)
5419.35i 1957 . 2 (∀𝑥𝜑 → ∃𝑥𝜓)
61, 5syl 17 1 (𝜑 → ∃𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1628  wex 1851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-6 2056
This theorem depends on definitions:  df-bi 197  df-ex 1852
This theorem is referenced by:  bj-spimevw  32988  bj-cbvexiw  32990
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