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Theorem spimv1 2101
 Description: Version of spim 2242 with a dv condition, which does not require ax-13 2234. See spimvw 1914 for a version with two dv conditions, requiring fewer axioms, and spimv 2245 for another variant. (Contributed by BJ, 31-May-2019.)
Hypotheses
Ref Expression
spimv1.nf 𝑥𝜓
spimv1.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spimv1 (∀𝑥𝜑𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem spimv1
StepHypRef Expression
1 spimv1.nf . 2 𝑥𝜓
2 ax6ev 1877 . . 3 𝑥 𝑥 = 𝑦
3 spimv1.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3eximii 1754 . 2 𝑥(𝜑𝜓)
51, 419.36i 2086 1 (∀𝑥𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701 This theorem is referenced by:  cbv3v  2158  bj-chvarv  31912  bj-cbv3v2  31914  wl-cbv3vv  32486
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