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Theorem reu6 3025
 Description: A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.)
Assertion
Ref Expression
reu6 (∃!x A φy A x A (φx = y))
Distinct variable groups:   x,y,A   φ,y
Allowed substitution hint:   φ(x)

Proof of Theorem reu6
StepHypRef Expression
1 df-reu 2621 . 2 (∃!x A φ∃!x(x A φ))
2 19.28v 1895 . . . . 5 (x(y A (x A → (φx = y))) ↔ (y A x(x A → (φx = y))))
3 eleq1 2413 . . . . . . . . . . . 12 (x = y → (x Ay A))
4 sbequ12 1919 . . . . . . . . . . . 12 (x = y → (φ ↔ [y / x]φ))
53, 4anbi12d 691 . . . . . . . . . . 11 (x = y → ((x A φ) ↔ (y A [y / x]φ)))
6 eqeq1 2359 . . . . . . . . . . 11 (x = y → (x = yy = y))
75, 6bibi12d 312 . . . . . . . . . 10 (x = y → (((x A φ) ↔ x = y) ↔ ((y A [y / x]φ) ↔ y = y)))
8 eqid 2353 . . . . . . . . . . . 12 y = y
98tbt 333 . . . . . . . . . . 11 ((y A [y / x]φ) ↔ ((y A [y / x]φ) ↔ y = y))
10 simpl 443 . . . . . . . . . . 11 ((y A [y / x]φ) → y A)
119, 10sylbir 204 . . . . . . . . . 10 (((y A [y / x]φ) ↔ y = y) → y A)
127, 11syl6bi 219 . . . . . . . . 9 (x = y → (((x A φ) ↔ x = y) → y A))
1312spimv 1990 . . . . . . . 8 (x((x A φ) ↔ x = y) → y A)
14 bi1 178 . . . . . . . . . . . 12 (((x A φ) ↔ x = y) → ((x A φ) → x = y))
1514expdimp 426 . . . . . . . . . . 11 ((((x A φ) ↔ x = y) x A) → (φx = y))
16 bi2 189 . . . . . . . . . . . . 13 (((x A φ) ↔ x = y) → (x = y → (x A φ)))
17 simpr 447 . . . . . . . . . . . . 13 ((x A φ) → φ)
1816, 17syl6 29 . . . . . . . . . . . 12 (((x A φ) ↔ x = y) → (x = yφ))
1918adantr 451 . . . . . . . . . . 11 ((((x A φ) ↔ x = y) x A) → (x = yφ))
2015, 19impbid 183 . . . . . . . . . 10 ((((x A φ) ↔ x = y) x A) → (φx = y))
2120ex 423 . . . . . . . . 9 (((x A φ) ↔ x = y) → (x A → (φx = y)))
2221sps 1754 . . . . . . . 8 (x((x A φ) ↔ x = y) → (x A → (φx = y)))
2313, 22jca 518 . . . . . . 7 (x((x A φ) ↔ x = y) → (y A (x A → (φx = y))))
2423a5i 1789 . . . . . 6 (x((x A φ) ↔ x = y) → x(y A (x A → (φx = y))))
25 bi1 178 . . . . . . . . . . 11 ((φx = y) → (φx = y))
2625imim2i 13 . . . . . . . . . 10 ((x A → (φx = y)) → (x A → (φx = y)))
2726imp3a 420 . . . . . . . . 9 ((x A → (φx = y)) → ((x A φ) → x = y))
2827adantl 452 . . . . . . . 8 ((y A (x A → (φx = y))) → ((x A φ) → x = y))
29 eleq1a 2422 . . . . . . . . . . . 12 (y A → (x = yx A))
3029adantr 451 . . . . . . . . . . 11 ((y A (x A → (φx = y))) → (x = yx A))
3130imp 418 . . . . . . . . . 10 (((y A (x A → (φx = y))) x = y) → x A)
32 bi2 189 . . . . . . . . . . . . . 14 ((φx = y) → (x = yφ))
3332imim2i 13 . . . . . . . . . . . . 13 ((x A → (φx = y)) → (x A → (x = yφ)))
3433com23 72 . . . . . . . . . . . 12 ((x A → (φx = y)) → (x = y → (x Aφ)))
3534imp 418 . . . . . . . . . . 11 (((x A → (φx = y)) x = y) → (x Aφ))
3635adantll 694 . . . . . . . . . 10 (((y A (x A → (φx = y))) x = y) → (x Aφ))
3731, 36jcai 522 . . . . . . . . 9 (((y A (x A → (φx = y))) x = y) → (x A φ))
3837ex 423 . . . . . . . 8 ((y A (x A → (φx = y))) → (x = y → (x A φ)))
3928, 38impbid 183 . . . . . . 7 ((y A (x A → (φx = y))) → ((x A φ) ↔ x = y))
4039alimi 1559 . . . . . 6 (x(y A (x A → (φx = y))) → x((x A φ) ↔ x = y))
4124, 40impbii 180 . . . . 5 (x((x A φ) ↔ x = y) ↔ x(y A (x A → (φx = y))))
42 df-ral 2619 . . . . . 6 (x A (φx = y) ↔ x(x A → (φx = y)))
4342anbi2i 675 . . . . 5 ((y A x A (φx = y)) ↔ (y A x(x A → (φx = y))))
442, 41, 433bitr4i 268 . . . 4 (x((x A φ) ↔ x = y) ↔ (y A x A (φx = y)))
4544exbii 1582 . . 3 (yx((x A φ) ↔ x = y) ↔ y(y A x A (φx = y)))
46 df-eu 2208 . . 3 (∃!x(x A φ) ↔ yx((x A φ) ↔ x = y))
47 df-rex 2620 . . 3 (y A x A (φx = y) ↔ y(y A x A (φx = y)))
4845, 46, 473bitr4i 268 . 2 (∃!x(x A φ) ↔ y A x A (φx = y))
491, 48bitri 240 1 (∃!x A φy A x A (φx = y))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∃!weu 2204  ∀wral 2614  ∃wrex 2615  ∃!wreu 2616 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-cleq 2346  df-clel 2349  df-ral 2619  df-rex 2620  df-reu 2621 This theorem is referenced by:  reu3  3026  reu6i  3027  reu8  3032
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