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Theorem spd 42223
Description: Specialization deduction, using implicit substitution. Based on the proof of spimed 2243. (Contributed by Emmett Weisz, 17-Jan-2020.)
Hypotheses
Ref Expression
spd.1 (𝜒 → Ⅎ𝑥𝜓)
spd.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spd (𝜒 → (∀𝑥𝜑𝜓))

Proof of Theorem spd
StepHypRef Expression
1 ax6e 2238 . . . 4 𝑥 𝑥 = 𝑦
2 spd.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
32biimpd 218 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 3eximii 1754 . . 3 𝑥(𝜑𝜓)
5419.35i 1795 . 2 (∀𝑥𝜑 → ∃𝑥𝜓)
6 spd.1 . . 3 (𝜒 → Ⅎ𝑥𝜓)
7619.9d 2058 . 2 (𝜒 → (∃𝑥𝜓𝜓))
85, 7syl5 33 1 (𝜒 → (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wal 1473  wex 1695  wnf 1699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701
This theorem is referenced by: (None)
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