MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  spimedv Structured version   Visualization version   GIF version

Theorem spimedv 2196
Description: Deduction version of spimev 2402. Version of spimed 2398 with a disjoint variable condition, which does not require ax-13 2382. See spime 2399 for a non-deduction version. (Contributed by NM, 14-May-1993.) (Revised by BJ, 31-May-2019.)
Hypotheses
Ref Expression
spimedv.1 (𝜒 → Ⅎ𝑥𝜑)
spimedv.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spimedv (𝜒 → (𝜑 → ∃𝑥𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem spimedv
StepHypRef Expression
1 spimedv.1 . . 3 (𝜒 → Ⅎ𝑥𝜑)
21nf5rd 2195 . 2 (𝜒 → (𝜑 → ∀𝑥𝜑))
3 ax6ev 1972 . . . 4 𝑥 𝑥 = 𝑦
4 spimedv.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4eximii 1838 . . 3 𝑥(𝜑𝜓)
6519.35i 1879 . 2 (∀𝑥𝜑 → ∃𝑥𝜓)
72, 6syl6 35 1 (𝜒 → (𝜑 → ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  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 2176
This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786
This theorem is referenced by:  spimefv  2197
  Copyright terms: Public domain W3C validator