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Theorem spv 2259
 Description: Specialization, using implicit substitution. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
spv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spv (∀𝑥𝜑𝜓)
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem spv
StepHypRef Expression
1 spv.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
21biimpd 219 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
32spimv 2256 1 (∀𝑥𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1478 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044  ax-13 2245 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702 This theorem is referenced by:  chvarv  2262  cbvalv  2272  ru  3421  nalset  4765  isowe2  6565  tfisi  7020  findcard2  8160  marypha1lem  8299  setind  8570  karden  8718  kmlem4  8935  axgroth3  9613  ramcl  15676  alexsubALTlem3  21793  i1fd  23388  dfpo2  31406  dfon2lem6  31447  trer  32005  axc11n-16  33742  elsetrecslem  41767
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