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Theorem spimed 1718
 Description: Deduction version of spime 1719. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.)
Hypotheses
Ref Expression
spimed.1 (𝜒 → Ⅎ𝑥𝜑)
spimed.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spimed (𝜒 → (𝜑 → ∃𝑥𝜓))

Proof of Theorem spimed
StepHypRef Expression
1 spimed.1 . . 3 (𝜒 → Ⅎ𝑥𝜑)
21nfrd 1500 . 2 (𝜒 → (𝜑 → ∀𝑥𝜑))
3 a9e 1674 . . . 4 𝑥 𝑥 = 𝑦
4 spimed.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4eximii 1581 . . 3 𝑥(𝜑𝜓)
6519.35i 1604 . 2 (∀𝑥𝜑 → ∃𝑥𝜓)
72, 6syl6 33 1 (𝜒 → (𝜑 → ∃𝑥𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1329  Ⅎwnf 1436  ∃wex 1468 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-i9 1510  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-nf 1437 This theorem is referenced by:  spime  1719
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