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Theorem spd 47911
Description: Specialization deduction, using implicit substitution. Based on the proof of spimed 2379. (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 2374 . . . 4 𝑥 𝑥 = 𝑦
2 spd.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
32biimpd 228 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 3eximii 1831 . . 3 𝑥(𝜑𝜓)
5419.35i 1873 . 2 (∀𝑥𝜑 → ∃𝑥𝜓)
6 spd.1 . . 3 (𝜒 → Ⅎ𝑥𝜓)
7619.9d 2188 . 2 (𝜒 → (∃𝑥𝜓𝜓))
85, 7syl5 34 1 (𝜒 → (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wex 1773  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163  ax-13 2363
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778
This theorem is referenced by: (None)
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