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Theorem amosym1 32102
Description: A symmetry with ∃*.

See negsym1 32093 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 2474 . 2 (∃*𝑥∃*𝑥⊥ ↔ (∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥))
2 mof 32086 . . . . 5 ∃*𝑥
3 19.8a 2049 . . . . . 6 (∃*𝑥⊥ → ∃𝑥∃*𝑥⊥)
43notnotd 138 . . . . 5 (∃*𝑥⊥ → ¬ ¬ ∃𝑥∃*𝑥⊥)
52, 4ax-mp 5 . . . 4 ¬ ¬ ∃𝑥∃*𝑥
65pm2.21i 116 . . 3 (¬ ∃𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
72notnoti 137 . . . . . 6 ¬ ¬ ∃*𝑥
87nex 1728 . . . . 5 ¬ ∃𝑥 ¬ ∃*𝑥
9 eunex 4824 . . . . 5 (∃!𝑥∃*𝑥⊥ → ∃𝑥 ¬ ∃*𝑥⊥)
108, 9mto 188 . . . 4 ¬ ∃!𝑥∃*𝑥
1110pm2.21i 116 . . 3 (∃!𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
126, 11ja 173 . 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 1485  wex 1701  ∃!weu 2469  ∃*wmo 2470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-nul 4754  ax-pow 4808
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-eu 2473  df-mo 2474
This theorem is referenced by: (None)
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