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Theorem eqreu 3624
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 3280 . . . . 5 (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ↔ (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ ∀𝑥𝐴 (𝑥 = 𝐵𝜑)))
2 eqreu.1 . . . . . . 7 (𝑥 = 𝐵 → (𝜑𝜓))
32ceqsralv 3452 . . . . . 6 (𝐵𝐴 → (∀𝑥𝐴 (𝑥 = 𝐵𝜑) ↔ 𝜓))
43anbi2d 624 . . . . 5 (𝐵𝐴 → ((∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ ∀𝑥𝐴 (𝑥 = 𝐵𝜑)) ↔ (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓)))
51, 4syl5bb 275 . . . 4 (𝐵𝐴 → (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ↔ (∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓)))
6 reu6i 3623 . . . . 5 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
76ex 403 . . . 4 (𝐵𝐴 → (∀𝑥𝐴 (𝜑𝑥 = 𝐵) → ∃!𝑥𝐴 𝜑))
85, 7sylbird 252 . . 3 (𝐵𝐴 → ((∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥𝐴 𝜑))
983impib 1150 . 2 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥𝐴 𝜑)
1093com23 1162 1 ((𝐵𝐴𝜓 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wral 3118  ∃!wreu 3120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2804
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-reu 3125  df-v 3417
This theorem is referenced by:  uzwo3  12067  frmdup3  17759  frgpup3  18545  neiptopreu  21309  ufileu  22094  mirreu  25977  lmireu  26100  symgfcoeu  30391
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