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Theorem eu1 2225
 Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypothesis
Ref Expression
eu1.1 yφ
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 nfs1v 2106 . . 3 x[y / x]φ
21euf 2210 . 2 (∃!y[y / x]φxy([y / x]φy = x))
3 eu1.1 . . 3 yφ
43sb8eu 2222 . 2 (∃!xφ∃!y[y / x]φ)
5 equcom 1680 . . . . . . 7 (x = yy = x)
65imbi2i 303 . . . . . 6 (([y / x]φx = y) ↔ ([y / x]φy = x))
76albii 1566 . . . . 5 (y([y / x]φx = y) ↔ y([y / x]φy = x))
83sb6rf 2091 . . . . 5 (φy(y = x → [y / x]φ))
97, 8anbi12i 678 . . . 4 ((y([y / x]φx = y) φ) ↔ (y([y / x]φy = x) y(y = x → [y / x]φ)))
10 ancom 437 . . . 4 ((φ y([y / x]φx = y)) ↔ (y([y / x]φx = y) φ))
11 albiim 1611 . . . 4 (y([y / x]φy = x) ↔ (y([y / x]φy = x) y(y = x → [y / x]φ)))
129, 10, 113bitr4i 268 . . 3 ((φ y([y / x]φx = y)) ↔ y([y / x]φy = x))
1312exbii 1582 . 2 (x(φ y([y / x]φx = y)) ↔ xy([y / x]φy = x))
142, 4, 133bitr4i 268 1 (∃!xφx(φ y([y / x]φx = y)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  [wsb 1648  ∃!weu 2204 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 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208 This theorem is referenced by:  euex  2227  eu2  2229  fvfullfunlem1  5861
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