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Theorem eqreu 3365
Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypothesis
Ref Expression
eqreu.1 (𝑥 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
eqreu ((𝐵𝐴𝜓 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eqreu
StepHypRef Expression
1 ralbiim 3051 . . . . 5 (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ↔ (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ ∀𝑥𝐴 (𝑥 = 𝐵𝜑)))
2 eqreu.1 . . . . . . 7 (𝑥 = 𝐵 → (𝜑𝜓))
32ceqsralv 3207 . . . . . 6 (𝐵𝐴 → (∀𝑥𝐴 (𝑥 = 𝐵𝜑) ↔ 𝜓))
43anbi2d 736 . . . . 5 (𝐵𝐴 → ((∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ ∀𝑥𝐴 (𝑥 = 𝐵𝜑)) ↔ (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓)))
51, 4syl5bb 271 . . . 4 (𝐵𝐴 → (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ↔ (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓)))
6 reu6i 3364 . . . . 5 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
76ex 449 . . . 4 (𝐵𝐴 → (∀𝑥𝐴 (𝜑𝑥 = 𝐵) → ∃!𝑥𝐴 𝜑))
85, 7sylbird 249 . . 3 (𝐵𝐴 → ((∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥𝐴 𝜑))
983impib 1254 . 2 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥𝐴 𝜑)
1093com23 1263 1 ((𝐵𝐴𝜓 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  ∃!wreu 2898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-v 3175
This theorem is referenced by:  uzwo3  11618  frmdup3  17176  frgpup3  17963  neiptopreu  20695  ufileu  21481  mirreu  25305  lmireu  25428  symgfcoeu  28970
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