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Theorem spv 2325
Description: Specialization, using implicit substitution. See spvv 1956 for a version using fewer axioms. (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 221 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
32spimv 2322 1 (∀𝑥𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-12 2107  ax-13 2302
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-nf 1748
This theorem is referenced by:  cbvalvOLD  2334  isowe2  6924  tfisi  7387  findcard2  8551  marypha1lem  8690  setind  8968  karden  9116  axgroth3  10049  ramcl  16219  alexsubALTlem3  22376  i1fd  24000  dfpo2  32548  dfon2lem6  32590  trer  33222  axc11n-16  35556  elsetrecslem  44202
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