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Theorem spv 1756
 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 136 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
32spimv 1708 1 (∀𝑥𝜑𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102  ∀wal 1257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443 This theorem depends on definitions:  df-bi 114  df-nf 1366 This theorem is referenced by:  chvarv  1828  ru  2786  nalset  3915  tfisi  4338  findcard2  6377  findcard2s  6378  bj-nalset  10402
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