Users' Mathboxes Mathbox for Anthony Hart < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  amosym1 Structured version   Visualization version   GIF version

Theorem amosym1 32550
Description: A symmetry with ∃*.

See negsym1 32541 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 2503 . 2 (∃*𝑥∃*𝑥⊥ ↔ (∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥))
2 mof 32534 . . . . 5 ∃*𝑥
3 19.8a 2090 . . . . . 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 1771 . . . . 5 ¬ ∃𝑥 ¬ ∃*𝑥
9 eunex 4889 . . . . 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 1528  wex 1744  ∃!weu 2498  ∃*wmo 2499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-nul 4822  ax-pow 4873
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-eu 2502  df-mo 2503
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator