Users' Mathboxes Mathbox for Emmett Weisz < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  spd Structured version   Visualization version   GIF version

Theorem spd 42953
Description: Specialization deduction, using implicit substitution. Based on the proof of spimed 2417. (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 2412 . . . 4 𝑥 𝑥 = 𝑦
2 spd.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
32biimpd 219 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 3eximii 1912 . . 3 𝑥(𝜑𝜓)
5419.35i 1958 . 2 (∀𝑥𝜑 → ∃𝑥𝜓)
6 spd.1 . . 3 (𝜒 → Ⅎ𝑥𝜓)
7619.9d 2225 . 2 (𝜒 → (∃𝑥𝜓𝜓))
85, 7syl5 34 1 (𝜒 → (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1629  wex 1852  wnf 1856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-nf 1858
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator