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Theorem spd 45147
Description: Specialization deduction, using implicit substitution. Based on the proof of spimed 2407. (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 2402 . . . 4 𝑥 𝑥 = 𝑦
2 spd.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
32biimpd 232 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 3eximii 1838 . . 3 𝑥(𝜑𝜓)
5419.35i 1879 . 2 (∀𝑥𝜑 → ∃𝑥𝜓)
6 spd.1 . . 3 (𝜒 → Ⅎ𝑥𝜓)
7619.9d 2204 . 2 (𝜒 → (∃𝑥𝜓𝜓))
85, 7syl5 34 1 (𝜒 → (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  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 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786
This theorem is referenced by: (None)
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