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Theorem eupickb 2269
 Description: Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
eupickb ((∃!xφ ∃!xψ x(φ ψ)) → (φψ))

Proof of Theorem eupickb
StepHypRef Expression
1 eupick 2267 . . 3 ((∃!xφ x(φ ψ)) → (φψ))
213adant2 974 . 2 ((∃!xφ ∃!xψ x(φ ψ)) → (φψ))
3 3simpc 954 . . 3 ((∃!xφ ∃!xψ x(φ ψ)) → (∃!xψ x(φ ψ)))
4 pm3.22 436 . . . . 5 ((φ ψ) → (ψ φ))
54eximi 1576 . . . 4 (x(φ ψ) → x(ψ φ))
65anim2i 552 . . 3 ((∃!xψ x(φ ψ)) → (∃!xψ x(ψ φ)))
7 eupick 2267 . . 3 ((∃!xψ x(ψ φ)) → (ψφ))
83, 6, 73syl 18 . 2 ((∃!xφ ∃!xψ x(φ ψ)) → (ψφ))
92, 8impbid 183 1 ((∃!xφ ∃!xψ x(φ ψ)) → (φψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541  ∃!weu 2204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209 This theorem is referenced by: (None)
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