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Theorem reu6i 3027
 Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
reu6i ((B A x A (φx = B)) → ∃!x A φ)
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem reu6i
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2362 . . . . 5 (y = B → (x = yx = B))
21bibi2d 309 . . . 4 (y = B → ((φx = y) ↔ (φx = B)))
32ralbidv 2634 . . 3 (y = B → (x A (φx = y) ↔ x A (φx = B)))
43rspcev 2955 . 2 ((B A x A (φx = B)) → y A x A (φx = y))
5 reu6 3025 . 2 (∃!x A φy A x A (φx = y))
64, 5sylibr 203 1 ((B A x A (φx = B)) → ∃!x A φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  ∃!wreu 2616 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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-reu 2621  df-v 2861 This theorem is referenced by:  eqreu  3028
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