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Theorem amosym1 32763
Description: A symmetry with ∃*.

See negsym1 32754 for more information. (Contributed by Anthony Hart, 13-Sep-2011.)

Assertion
Ref Expression
amosym1 (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)

Proof of Theorem amosym1
StepHypRef Expression
1 df-mo 2623 . 2 (∃*𝑥∃*𝑥⊥ ↔ (∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥))
2 mof 32747 . . . . 5 ∃*𝑥
3 19.8a 2206 . . . . . 6 (∃*𝑥⊥ → ∃𝑥∃*𝑥⊥)
43notnotd 140 . . . . 5 (∃*𝑥⊥ → ¬ ¬ ∃𝑥∃*𝑥⊥)
52, 4ax-mp 5 . . . 4 ¬ ¬ ∃𝑥∃*𝑥
65pm2.21i 117 . . 3 (¬ ∃𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
72notnoti 139 . . . . . 6 ¬ ¬ ∃*𝑥
87nex 1879 . . . . 5 ¬ ∃𝑥 ¬ ∃*𝑥
9 eunex 4991 . . . . 5 (∃!𝑥∃*𝑥⊥ → ∃𝑥 ¬ ∃*𝑥⊥)
108, 9mto 188 . . . 4 ¬ ∃!𝑥∃*𝑥
1110pm2.21i 117 . . 3 (∃!𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
126, 11ja 174 . 2 ((∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥) → ∃*𝑥𝜑)
131, 12sylbi 207 1 (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wfal 1636  wex 1852  ∃!weu 2618  ∃*wmo 2619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-nul 4924  ax-pow 4975
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-eu 2622  df-mo 2623
This theorem is referenced by: (None)
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