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Theorem eqreu 2755
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 2464 . . . . 5 (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ↔ (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ ∀𝑥𝐴 (𝑥 = 𝐵𝜑)))
2 eqreu.1 . . . . . . 7 (𝑥 = 𝐵 → (𝜑𝜓))
32ceqsralv 2602 . . . . . 6 (𝐵𝐴 → (∀𝑥𝐴 (𝑥 = 𝐵𝜑) ↔ 𝜓))
43anbi2d 445 . . . . 5 (𝐵𝐴 → ((∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ ∀𝑥𝐴 (𝑥 = 𝐵𝜑)) ↔ (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓)))
51, 4syl5bb 185 . . . 4 (𝐵𝐴 → (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ↔ (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓)))
6 reu6i 2754 . . . . 5 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
76ex 112 . . . 4 (𝐵𝐴 → (∀𝑥𝐴 (𝜑𝑥 = 𝐵) → ∃!𝑥𝐴 𝜑))
85, 7sylbird 163 . . 3 (𝐵𝐴 → ((∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥𝐴 𝜑))
983impib 1113 . 2 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥𝐴 𝜑)
1093com23 1121 1 ((𝐵𝐴𝜓 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  wral 2323  ∃!wreu 2325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-v 2576
This theorem is referenced by: (None)
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