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Theorem eupickbi 2270
 Description: Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
eupickbi (∃!xφ → (x(φ ψ) ↔ x(φψ)))

Proof of Theorem eupickbi
StepHypRef Expression
1 eupicka 2268 . . 3 ((∃!xφ x(φ ψ)) → x(φψ))
21ex 423 . 2 (∃!xφ → (x(φ ψ) → x(φψ)))
3 nfa1 1788 . . . . 5 xx(φψ)
4 ancl 529 . . . . . . 7 ((φψ) → (φ → (φ ψ)))
5 simpl 443 . . . . . . 7 ((φ ψ) → φ)
64, 5impbid1 194 . . . . . 6 ((φψ) → (φ ↔ (φ ψ)))
76sps 1754 . . . . 5 (x(φψ) → (φ ↔ (φ ψ)))
83, 7eubid 2211 . . . 4 (x(φψ) → (∃!xφ∃!x(φ ψ)))
9 euex 2227 . . . 4 (∃!x(φ ψ) → x(φ ψ))
108, 9syl6bi 219 . . 3 (x(φψ) → (∃!xφx(φ ψ)))
1110com12 27 . 2 (∃!xφ → (x(φψ) → x(φ ψ)))
122, 11impbid 183 1 (∃!xφ → (x(φ ψ) ↔ x(φψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃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-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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