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Theorem eqreu 3028
 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 ((B A ψ x A (φx = B)) → ∃!x A φ)
Distinct variable groups:   x,A   x,B   ψ,x
Allowed substitution hint:   φ(x)

Proof of Theorem eqreu
StepHypRef Expression
1 ralbiim 2751 . . . . 5 (x A (φx = B) ↔ (x A (φx = B) x A (x = Bφ)))
2 eqreu.1 . . . . . . 7 (x = B → (φψ))
32ceqsralv 2886 . . . . . 6 (B A → (x A (x = Bφ) ↔ ψ))
43anbi2d 684 . . . . 5 (B A → ((x A (φx = B) x A (x = Bφ)) ↔ (x A (φx = B) ψ)))
51, 4syl5bb 248 . . . 4 (B A → (x A (φx = B) ↔ (x A (φx = B) ψ)))
6 reu6i 3027 . . . . 5 ((B A x A (φx = B)) → ∃!x A φ)
76ex 423 . . . 4 (B A → (x A (φx = B) → ∃!x A φ))
85, 7sylbird 226 . . 3 (B A → ((x A (φx = B) ψ) → ∃!x A φ))
983impib 1149 . 2 ((B A x A (φx = B) ψ) → ∃!x A φ)
1093com23 1157 1 ((B A ψ x A (φx = B)) → ∃!x A φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃!wreu 2616 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-reu 2621  df-v 2861 This theorem is referenced by: (None)
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