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Theorem eu1 1903
Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.)
Hypothesis
Ref Expression
eu1.1 (φyφ)
Assertion
Ref Expression
eu1 (∃!xφx(φ y([y / x]φx = y)))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem eu1
StepHypRef Expression
1 hbs1 1792 . . 3 ([y / x]φx[y / x]φ)
21euf 1883 . 2 (∃!y[y / x]φxy([y / x]φy = x))
3 eu1.1 . . 3 (φyφ)
43sb8euh 1901 . 2 (∃!xφ∃!y[y / x]φ)
5 equcom 1571 . . . . . . 7 (x = yy = x)
65imbi2i 215 . . . . . 6 (([y / x]φx = y) ↔ ([y / x]φy = x))
76albii 1335 . . . . 5 (y([y / x]φx = y) ↔ y([y / x]φy = x))
83sb6rf 1711 . . . . 5 (φy(y = x → [y / x]φ))
97, 8anbi12i 436 . . . 4 ((y([y / x]φx = y) φ) ↔ (y([y / x]φy = x) y(y = x → [y / x]φ)))
10 ancom 253 . . . 4 ((φ y([y / x]φx = y)) ↔ (y([y / x]φx = y) φ))
11 albiim 1353 . . . 4 (y([y / x]φy = x) ↔ (y([y / x]φy = x) y(y = x → [y / x]φ)))
129, 10, 113bitr4i 201 . . 3 ((φ y([y / x]φx = y)) ↔ y([y / x]φy = x))
1312exbii 1474 . 2 (x(φ y([y / x]φx = y)) ↔ xy([y / x]φy = x))
142, 4, 133bitr4i 201 1 (∃!xφx(φ y([y / x]φx = y)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1224  wex 1358  [wsb 1623  ∃!weu 1878
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-eu 1881
This theorem is referenced by:  euex  1908  eu2  1922
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