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Theorem amosym1 31595
Description: A symmetry with ∃*.

See negsym1 31586 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 2463 . 2 (∃*𝑥∃*𝑥⊥ ↔ (∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥))
2 mof 31579 . . . . 5 ∃*𝑥
3 19.8a 2039 . . . . . 6 (∃*𝑥⊥ → ∃𝑥∃*𝑥⊥)
43notnotd 137 . . . . 5 (∃*𝑥⊥ → ¬ ¬ ∃𝑥∃*𝑥⊥)
52, 4ax-mp 5 . . . 4 ¬ ¬ ∃𝑥∃*𝑥
65pm2.21i 115 . . 3 (¬ ∃𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
72notnoti 136 . . . . . 6 ¬ ¬ ∃*𝑥
87nex 1722 . . . . 5 ¬ ∃𝑥 ¬ ∃*𝑥
9 eunex 4785 . . . . 5 (∃!𝑥∃*𝑥⊥ → ∃𝑥 ¬ ∃*𝑥⊥)
108, 9mto 187 . . . 4 ¬ ∃!𝑥∃*𝑥
1110pm2.21i 115 . . 3 (∃!𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
126, 11ja 172 . 2 ((∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥) → ∃*𝑥𝜑)
131, 12sylbi 206 1 (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wfal 1480  wex 1695  ∃!weu 2458  ∃*wmo 2459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-nul 4717  ax-pow 4769
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463
This theorem is referenced by: (None)
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