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Theorem eu4 1959
Description: Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
Hypothesis
Ref Expression
eu4.1 (x = y → (φψ))
Assertion
Ref Expression
eu4 (∃!xφ ↔ (xφ xy((φ ψ) → x = y)))
Distinct variable groups:   x,y   φ,y   ψ,x
Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem eu4
StepHypRef Expression
1 eu5 1944 . 2 (∃!xφ ↔ (xφ ∃*xφ))
2 eu4.1 . . . 4 (x = y → (φψ))
32mo4 1958 . . 3 (∃*xφxy((φ ψ) → x = y))
43anbi2i 430 . 2 ((xφ ∃*xφ) ↔ (xφ xy((φ ψ) → x = y)))
51, 4bitri 173 1 (∃!xφ ↔ (xφ xy((φ ψ) → x = y)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wex 1378  ∃!weu 1897  ∃*wmo 1898
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
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901
This theorem is referenced by:  euequ1  1992  eueq  2706  euind  2722  eusv1  4150  eroveu  6133
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