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Theorem exists2OLD 2694
 Description: Obsolete version of exists2 2693 as of 4-Mar-2023. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
exists2OLD ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)

Proof of Theorem exists2OLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfeu1 2609 . . . . . 6 𝑥∃!𝑥 𝑥 = 𝑥
2 nfa1 2145 . . . . . 6 𝑥𝑥𝜑
3 exists1 2692 . . . . . . 7 (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
4 axc16 2234 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
53, 4sylbi 209 . . . . . 6 (∃!𝑥 𝑥 = 𝑥 → (𝜑 → ∀𝑥𝜑))
61, 2, 5exlimd 2204 . . . . 5 (∃!𝑥 𝑥 = 𝑥 → (∃𝑥𝜑 → ∀𝑥𝜑))
76com12 32 . . . 4 (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ∀𝑥𝜑))
8 alex 1869 . . . 4 (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
97, 8syl6ib 243 . . 3 (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ¬ ∃𝑥 ¬ 𝜑))
109con2d 132 . 2 (∃𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ¬ ∃!𝑥 𝑥 = 𝑥))
1110imp 397 1 ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386  ∀wal 1599  ∃wex 1823  ∃!weu 2586 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-mo 2551  df-eu 2587 This theorem is referenced by: (None)
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