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Theorem spimed 2395
Description: Deduction version of spime 2396. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker spimedv 2195 if possible. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) (New usage is discouraged.)
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)
21nf5rd 2194 . 2 (𝜒 → (𝜑 → ∀𝑥𝜑))
3 ax6e 2390 . . . 4 𝑥 𝑥 = 𝑦
4 spimed.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4eximii 1838 . . 3 𝑥(𝜑𝜓)
6519.35i 1879 . 2 (∀𝑥𝜑 → ∃𝑥𝜓)
72, 6syl6 35 1 (𝜒 → (𝜑 → ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wex 1781  wnf 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175  ax-13 2379
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786
This theorem is referenced by:  spime  2396  2ax6elem  2482
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