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Theorem eqeu 3682
Description: A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
Hypothesis
Ref Expression
eqeu.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
eqeu ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃!𝑥𝜑)
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem eqeu
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqeu.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
21spcegv 3582 . . . 4 (𝐴𝐵 → (𝜓 → ∃𝑥𝜑))
32imp 410 . . 3 ((𝐴𝐵𝜓) → ∃𝑥𝜑)
433adant3 1129 . 2 ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃𝑥𝜑)
5 eqeq2 2836 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
65imbi2d 344 . . . . . 6 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
76albidv 1922 . . . . 5 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
87spcegv 3582 . . . 4 (𝐴𝐵 → (∀𝑥(𝜑𝑥 = 𝐴) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
98imp 410 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
1093adant2 1128 . 2 ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
11 eu3v 2656 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
124, 10, 11sylanbrc 586 1 ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084  wal 1536   = wceq 1538  wex 1781  wcel 2115  ∃!weu 2654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-mo 2624  df-eu 2655  df-cleq 2817  df-clel 2896
This theorem is referenced by:  rngurd  30875  neibastop3  33728  upixp  35067  zrinitorngc  44466  zrtermorngc  44467  zrtermoringc  44536
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