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Theorem spv 2243
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 217 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
32spimv 2240 1 (∀𝑥𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2032  ax-13 2229
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by:  chvarv  2246  cbvalv  2256  ru  3397  nalset  4715  isowe2  6475  tfisi  6924  findcard2  8059  marypha1lem  8196  setind  8467  karden  8615  kmlem4  8832  axgroth3  9506  ramcl  15514  alexsubALTlem3  21602  i1fd  23168  dfpo2  30701  dfon2lem6  30740  trer  31283  axc11n-16  33041
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