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Theorem rmo2 3131
 Description: Alternate definition of restricted "at most one." Note that ∃*x ∈ Aφ is not equivalent to ∃y ∈ A∀x ∈ A(φ → x = y) (in analogy to reu6 3025); to see this, let A be the empty set. However, one direction of this pattern holds; see rmo2i 3132. (Contributed by NM, 17-Jun-2017.)
Hypothesis
Ref Expression
rmo2.1 yφ
Assertion
Ref Expression
rmo2 (∃*x A φyx A (φx = y))
Distinct variable group:   x,y,A
Allowed substitution hints:   φ(x,y)

Proof of Theorem rmo2
StepHypRef Expression
1 df-rmo 2622 . 2 (∃*x A φ∃*x(x A φ))
2 nfv 1619 . . . 4 y x A
3 rmo2.1 . . . 4 yφ
42, 3nfan 1824 . . 3 y(x A φ)
54mo2 2233 . 2 (∃*x(x A φ) ↔ yx((x A φ) → x = y))
6 impexp 433 . . . . 5 (((x A φ) → x = y) ↔ (x A → (φx = y)))
76albii 1566 . . . 4 (x((x A φ) → x = y) ↔ x(x A → (φx = y)))
8 df-ral 2619 . . . 4 (x A (φx = y) ↔ x(x A → (φx = y)))
97, 8bitr4i 243 . . 3 (x((x A φ) → x = y) ↔ x A (φx = y))
109exbii 1582 . 2 (yx((x A φ) → x = y) ↔ yx A (φx = y))
111, 5, 103bitri 262 1 (∃*x A φyx A (φx = y))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   ∈ wcel 1710  ∃*wmo 2205  ∀wral 2614  ∃*wrmo 2617 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  df-ral 2619  df-rmo 2622 This theorem is referenced by:  rmo2i  3132
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