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Theorem reu6 2701
 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 2285 . 2 (∃!x A φ∃!x(x A φ))
2 19.28v 1756 . . . . 5 (x(y A (x A → (φx = y))) ↔ (y A x(x A → (φx = y))))
3 eleq1 2076 . . . . . . . . . . . 12 (x = y → (x Ay A))
4 sbequ12 1630 . . . . . . . . . . . 12 (x = y → (φ ↔ [y / x]φ))
53, 4anbi12d 442 . . . . . . . . . . 11 (x = y → ((x A φ) ↔ (y A [y / x]φ)))
6 equequ1 1574 . . . . . . . . . . 11 (x = y → (x = yy = y))
75, 6bibi12d 224 . . . . . . . . . 10 (x = y → (((x A φ) ↔ x = y) ↔ ((y A [y / x]φ) ↔ y = y)))
8 equid 1565 . . . . . . . . . . . 12 y = y
98tbt 236 . . . . . . . . . . 11 ((y A [y / x]φ) ↔ ((y A [y / x]φ) ↔ y = y))
10 simpl 102 . . . . . . . . . . 11 ((y A [y / x]φ) → y A)
119, 10sylbir 125 . . . . . . . . . 10 (((y A [y / x]φ) ↔ y = y) → y A)
127, 11syl6bi 152 . . . . . . . . 9 (x = y → (((x A φ) ↔ x = y) → y A))
1312spimv 1668 . . . . . . . 8 (x((x A φ) ↔ x = y) → y A)
14 bi1 111 . . . . . . . . . . . 12 (((x A φ) ↔ x = y) → ((x A φ) → x = y))
1514expdimp 246 . . . . . . . . . . 11 ((((x A φ) ↔ x = y) x A) → (φx = y))
16 bi2 121 . . . . . . . . . . . . 13 (((x A φ) ↔ x = y) → (x = y → (x A φ)))
17 simpr 103 . . . . . . . . . . . . 13 ((x A φ) → φ)
1816, 17syl6 29 . . . . . . . . . . . 12 (((x A φ) ↔ x = y) → (x = yφ))
1918adantr 261 . . . . . . . . . . 11 ((((x A φ) ↔ x = y) x A) → (x = yφ))
2015, 19impbid 120 . . . . . . . . . 10 ((((x A φ) ↔ x = y) x A) → (φx = y))
2120ex 108 . . . . . . . . 9 (((x A φ) ↔ x = y) → (x A → (φx = y)))
2221sps 1406 . . . . . . . 8 (x((x A φ) ↔ x = y) → (x A → (φx = y)))
2313, 22jca 290 . . . . . . 7 (x((x A φ) ↔ x = y) → (y A (x A → (φx = y))))
2423a5i 1411 . . . . . 6 (x((x A φ) ↔ x = y) → x(y A (x A → (φx = y))))
25 bi1 111 . . . . . . . . . . 11 ((φx = y) → (φx = y))
2625imim2i 12 . . . . . . . . . 10 ((x A → (φx = y)) → (x A → (φx = y)))
2726impd 242 . . . . . . . . 9 ((x A → (φx = y)) → ((x A φ) → x = y))
2827adantl 262 . . . . . . . 8 ((y A (x A → (φx = y))) → ((x A φ) → x = y))
29 eleq1a 2085 . . . . . . . . . . . 12 (y A → (x = yx A))
3029adantr 261 . . . . . . . . . . 11 ((y A (x A → (φx = y))) → (x = yx A))
3130imp 115 . . . . . . . . . 10 (((y A (x A → (φx = y))) x = y) → x A)
32 bi2 121 . . . . . . . . . . . . . 14 ((φx = y) → (x = yφ))
3332imim2i 12 . . . . . . . . . . . . 13 ((x A → (φx = y)) → (x A → (x = yφ)))
3433com23 72 . . . . . . . . . . . 12 ((x A → (φx = y)) → (x = y → (x Aφ)))
3534imp 115 . . . . . . . . . . 11 (((x A → (φx = y)) x = y) → (x Aφ))
3635adantll 445 . . . . . . . . . 10 (((y A (x A → (φx = y))) x = y) → (x Aφ))
3731, 36jcai 294 . . . . . . . . 9 (((y A (x A → (φx = y))) x = y) → (x A φ))
3837ex 108 . . . . . . . 8 ((y A (x A → (φx = y))) → (x = y → (x A φ)))
3928, 38impbid 120 . . . . . . 7 ((y A (x A → (φx = y))) → ((x A φ) ↔ x = y))
4039alimi 1320 . . . . . 6 (x(y A (x A → (φx = y))) → x((x A φ) ↔ x = y))
4124, 40impbii 117 . . . . 5 (x((x A φ) ↔ x = y) ↔ x(y A (x A → (φx = y))))
42 df-ral 2283 . . . . . 6 (x A (φx = y) ↔ x(x A → (φx = y)))
4342anbi2i 430 . . . . 5 ((y A x A (φx = y)) ↔ (y A x(x A → (φx = y))))
442, 41, 433bitr4i 201 . . . 4 (x((x A φ) ↔ x = y) ↔ (y A x A (φx = y)))
4544exbii 1472 . . 3 (yx((x A φ) ↔ x = y) ↔ y(y A x A (φx = y)))
46 df-eu 1879 . . 3 (∃!x(x A φ) ↔ yx((x A φ) ↔ x = y))
47 df-rex 2284 . . 3 (y A x A (φx = y) ↔ y(y A x A (φx = y)))
4845, 46, 473bitr4i 201 . 2 (∃!x(x A φ) ↔ y A x A (φx = y))
491, 48bitri 173 1 (∃!x A φy A x A (φx = y))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1224  ∃wex 1357   ∈ wcel 1369  [wsb 1621  ∃!weu 1876  ∀wral 2278  ∃wrex 2279  ∃!wreu 2280 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-5 1312  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-4 1376  ax-17 1395  ax-i9 1399  ax-ial 1403  ax-ext 1998 This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1622  df-eu 1879  df-cleq 2009  df-clel 2012  df-ral 2283  df-rex 2284  df-reu 2285 This theorem is referenced by:  reu3  2702  reu6i  2703  reu8  2708
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