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Theorem spv 1882
Description: Specialization, using implicit substitition. (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 144 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
32spimv 1833 1 (∀𝑥𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556
This theorem depends on definitions:  df-bi 117  df-nf 1483
This theorem is referenced by:  spvv  1930  cbvalvw  1942  chvarv  1964  ru  2996  nalset  4173  tfisi  4634  tfr1onlemsucfn  6425  tfr1onlemsucaccv  6426  tfr1onlembxssdm  6428  tfr1onlembfn  6429  tfr1onlemres  6434  tfri1dALT  6436  tfrcllemsucfn  6438  tfrcllemsucaccv  6439  tfrcllembxssdm  6441  tfrcllembfn  6442  tfrcllemres  6447  findcard2  6985  findcard2s  6986  bj-nalset  15764
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