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Theorem spimed 1675
Description: Deduction version of spime 1676. (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 1458 . 2 (𝜒 → (𝜑 → ∀𝑥𝜑))
3 a9e 1631 . . . 4 𝑥 𝑥 = 𝑦
4 spimed.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4eximii 1538 . . 3 𝑥(𝜑𝜓)
6519.35i 1561 . 2 (∀𝑥𝜑 → ∃𝑥𝜓)
72, 6syl6 33 1 (𝜒 → (𝜑 → ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  wnf 1394  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  spime  1676
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