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Theorem bj-spimev 33218
Description: Version of spime 2396 with a disjoint variable condition, which does not require ax-13 2377. (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 33217 . 2 (⊤ → (𝜑 → ∃𝑥𝜓))
54mptru 1661 1 (𝜑 → ∃𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wtru 1654  wex 1875  wnf 1879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213
This theorem depends on definitions:  df-bi 199  df-tru 1657  df-ex 1876  df-nf 1880
This theorem is referenced by:  bj-spimevv  33220
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