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Theorem eqreu 2794
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 2496 . . . . 5 (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ↔ (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ ∀𝑥𝐴 (𝑥 = 𝐵𝜑)))
2 eqreu.1 . . . . . . 7 (𝑥 = 𝐵 → (𝜑𝜓))
32ceqsralv 2639 . . . . . 6 (𝐵𝐴 → (∀𝑥𝐴 (𝑥 = 𝐵𝜑) ↔ 𝜓))
43anbi2d 452 . . . . 5 (𝐵𝐴 → ((∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ ∀𝑥𝐴 (𝑥 = 𝐵𝜑)) ↔ (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓)))
51, 4syl5bb 190 . . . 4 (𝐵𝐴 → (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ↔ (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓)))
6 reu6i 2793 . . . . 5 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
76ex 113 . . . 4 (𝐵𝐴 → (∀𝑥𝐴 (𝜑𝑥 = 𝐵) → ∃!𝑥𝐴 𝜑))
85, 7sylbird 168 . . 3 (𝐵𝐴 → ((∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥𝐴 𝜑))
983impib 1137 . 2 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥𝐴 𝜑)
1093com23 1145 1 ((𝐵𝐴𝜓 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wcel 1434  wral 2353  ∃!wreu 2355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-v 2612
This theorem is referenced by: (None)
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