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Theorem amosym1 33331
Description: A symmetry with ∃*.

See negsym1 33322 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 2603 . 2 (∃*𝑥∃*𝑥⊥ ↔ (∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥))
2 mofal 33315 . . . . 5 ∃*𝑥
3 19.8a 2110 . . . . . 6 (∃*𝑥⊥ → ∃𝑥∃*𝑥⊥)
43notnotd 141 . . . . 5 (∃*𝑥⊥ → ¬ ¬ ∃𝑥∃*𝑥⊥)
52, 4ax-mp 5 . . . 4 ¬ ¬ ∃𝑥∃*𝑥
65pm2.21i 117 . . 3 (¬ ∃𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
72notnoti 140 . . . . . 6 ¬ ¬ ∃*𝑥
87nex 1764 . . . . 5 ¬ ∃𝑥 ¬ ∃*𝑥
9 eunex 5139 . . . . 5 (∃!𝑥∃*𝑥⊥ → ∃𝑥 ¬ ∃*𝑥⊥)
108, 9mto 189 . . . 4 ¬ ∃!𝑥∃*𝑥
1110pm2.21i 117 . . 3 (∃!𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
126, 11ja 175 . 2 ((∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥) → ∃*𝑥𝜑)
131, 12sylbi 209 1 (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wfal 1520  wex 1743  ∃*wmo 2546  ∃!weu 2584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-12 2107  ax-nul 5063  ax-pow 5115
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-mo 2548  df-eu 2585
This theorem is referenced by: (None)
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