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Theorem amosym1 33848
Description: A symmetry with ∃*.

See negsym1 33839 for more information. (Contributed by Anthony Hart, 13-Sep-2011.)

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

Proof of Theorem amosym1
StepHypRef Expression
1 moeu 2667 . 2 (∃*𝑥∃*𝑥⊥ ↔ (∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥))
2 mofal 33831 . . . . 5 ∃*𝑥
3 19.8a 2181 . . . . . 6 (∃*𝑥⊥ → ∃𝑥∃*𝑥⊥)
43notnotd 146 . . . . 5 (∃*𝑥⊥ → ¬ ¬ ∃𝑥∃*𝑥⊥)
52, 4ax-mp 5 . . . 4 ¬ ¬ ∃𝑥∃*𝑥
65pm2.21i 119 . . 3 (¬ ∃𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
72notnoti 145 . . . . . 6 ¬ ¬ ∃*𝑥
87nex 1802 . . . . 5 ¬ ∃𝑥 ¬ ∃*𝑥
9 eunex 5268 . . . . 5 (∃!𝑥∃*𝑥⊥ → ∃𝑥 ¬ ∃*𝑥⊥)
108, 9mto 200 . . . 4 ¬ ∃!𝑥∃*𝑥
1110pm2.21i 119 . . 3 (∃!𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
126, 11ja 189 . 2 ((∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥) → ∃*𝑥𝜑)
131, 12sylbi 220 1 (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wfal 1550  wex 1781  ∃*wmo 2620  ∃!weu 2652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2178  ax-nul 5186  ax-pow 5243
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-mo 2622  df-eu 2653
This theorem is referenced by: (None)
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