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Theorem spimv 2397
 Description: A version of spim 2394 with a distinct variable requirement instead of a bound-variable hypothesis. Usage of this theorem is discouraged because it depends on ax-13 2379. See spimfv 2239 and spimvw 2002 for versions requiring fewer axioms. (Contributed by NM, 31-Jul-1993.) (New usage is discouraged.)
Hypothesis
Ref Expression
spimv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spimv (∀𝑥𝜑𝜓)
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem spimv
StepHypRef Expression
1 nfv 1915 . 2 𝑥𝜓
2 spimv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2spim 2394 1 (∀𝑥𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786 This theorem is referenced by:  spv  2400
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