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Theorem ax12ev2 2176
Description: Version of ax12v2 2175 rewritten to use an existential quantifier. One direction of sbalex 2238 without the universal quantifier, avoiding ax-10 2136. (Contributed by SN, 14-Aug-2025.)
Assertion
Ref Expression
ax12ev2 (∃𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax12ev2
StepHypRef Expression
1 exnalimn 1842 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))
2 ax12v2 2175 . . . 4 (𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
32con1d 145 . . 3 (𝑥 = 𝑦 → (¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) → 𝜑))
41, 3biimtrid 242 . 2 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑))
54com12 32 1 (∃𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535  wex 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2173
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778
This theorem is referenced by:  sbalex  2238  mopick  2622  sbalexi  42155
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