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Theorem spimed 2397
Description: Deduction version of spime 2398. See also spimedv 2187. (Contributed by NM, 14-May-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)
21nf5rd 2186 . 2 (𝜒 → (𝜑 → ∀𝑥𝜑))
3 ax6e 2392 . . . 4 𝑥 𝑥 = 𝑦
4 spimed.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4eximii 1828 . . 3 𝑥(𝜑𝜓)
6519.35i 1870 . 2 (∀𝑥𝜑 → ∃𝑥𝜓)
72, 6syl6 35 1 (𝜒 → (𝜑 → ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526  wex 1771  wnf 1775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776
This theorem is referenced by:  spime  2398  2ax6elem  2485
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