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Theorem bj-spimev 31907
 Description: Version of spime 2244 with a dv condition, which does not require ax-13 2234. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-spimev.1 𝑥𝜑
bj-spimev.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-spimev (𝜑 → ∃𝑥𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-spimev
StepHypRef Expression
1 bj-spimev.1 . . . 4 𝑥𝜑
21a1i 11 . . 3 (⊤ → Ⅎ𝑥𝜑)
3 bj-spimev.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3bj-spimedv 31906 . 2 (⊤ → (𝜑 → ∃𝑥𝜓))
54trud 1484 1 (𝜑 → ∃𝑥𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ⊤wtru 1476  ∃wex 1695  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-tru 1478  df-ex 1696  df-nf 1701 This theorem is referenced by:  bj-spimevv  31909
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