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Theorem reu4 3030
 Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.)
Hypothesis
Ref Expression
rmo4.1 (x = y → (φψ))
Assertion
Ref Expression
reu4 (∃!x A φ ↔ (x A φ x A y A ((φ ψ) → x = y)))
Distinct variable groups:   x,y,A   φ,y   ψ,x
Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem reu4
StepHypRef Expression
1 reu5 2824 . 2 (∃!x A φ ↔ (x A φ ∃*x A φ))
2 rmo4.1 . . . 4 (x = y → (φψ))
32rmo4 3029 . . 3 (∃*x A φx A y A ((φ ψ) → x = y))
43anbi2i 675 . 2 ((x A φ ∃*x A φ) ↔ (x A φ x A y A ((φ ψ) → x = y)))
51, 4bitri 240 1 (∃!x A φ ↔ (x A φ x A y A ((φ ψ) → x = y)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wral 2614  ∃wrex 2615  ∃!wreu 2616  ∃*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  ax-ext 2334 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-cleq 2346  df-clel 2349  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622 This theorem is referenced by:  reuind  3039  vfinnc  4471  nnpw1ex  4484  ncspw1eu  6159
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