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Theorem spd 42190
 Description: Specialization deduction, using implicit substitution. Based on the proof of spimed 2253. (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 2248 . . . 4 𝑥 𝑥 = 𝑦
2 spd.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
32biimpd 219 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 3eximii 1762 . . 3 𝑥(𝜑𝜓)
5419.35i 1804 . 2 (∀𝑥𝜑 → ∃𝑥𝜓)
6 spd.1 . . 3 (𝜒 → Ⅎ𝑥𝜓)
7619.9d 2068 . 2 (𝜒 → (∃𝑥𝜓𝜓))
85, 7syl5 34 1 (𝜒 → (∀𝑥𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1479  ∃wex 1702  Ⅎwnf 1706 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045  ax-13 2244 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703  df-nf 1708 This theorem is referenced by: (None)
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