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Theorem eqreu 2727
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 2441 . . . . 5 (x A (φx = B) ↔ (x A (φx = B) x A (x = Bφ)))
2 eqreu.1 . . . . . . 7 (x = B → (φψ))
32ceqsralv 2579 . . . . . 6 (B A → (x A (x = Bφ) ↔ ψ))
43anbi2d 437 . . . . 5 (B A → ((x A (φx = B) x A (x = Bφ)) ↔ (x A (φx = B) ψ)))
51, 4syl5bb 181 . . . 4 (B A → (x A (φx = B) ↔ (x A (φx = B) ψ)))
6 reu6i 2726 . . . . 5 ((B A x A (φx = B)) → ∃!x A φ)
76ex 108 . . . 4 (B A → (x A (φx = B) → ∃!x A φ))
85, 7sylbird 159 . . 3 (B A → ((x A (φx = B) ψ) → ∃!x A φ))
983impib 1101 . 2 ((B A x A (φx = B) ψ) → ∃!x A φ)
1093com23 1109 1 ((B A ψ x A (φx = B)) → ∃!x A φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  wral 2300  ∃!wreu 2302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553
This theorem is referenced by: (None)
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