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Theorem iotaval 4350
 Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotaval (x(φx = y) → (℩xφ) = y)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem iotaval
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 dfiota2 4340 . 2 (℩xφ) = {z x(φx = z)}
2 vex 2862 . . . . . . 7 y V
3 sbeqalb 3098 . . . . . . . 8 (y V → ((x(φx = y) x(φx = z)) → y = z))
4 equcomi 1679 . . . . . . . 8 (y = zz = y)
53, 4syl6 29 . . . . . . 7 (y V → ((x(φx = y) x(φx = z)) → z = y))
62, 5ax-mp 8 . . . . . 6 ((x(φx = y) x(φx = z)) → z = y)
76ex 423 . . . . 5 (x(φx = y) → (x(φx = z) → z = y))
8 equequ2 1686 . . . . . . . . . 10 (y = z → (x = yx = z))
98eqcoms 2356 . . . . . . . . 9 (z = y → (x = yx = z))
109bibi2d 309 . . . . . . . 8 (z = y → ((φx = y) ↔ (φx = z)))
1110biimpd 198 . . . . . . 7 (z = y → ((φx = y) → (φx = z)))
1211alimdv 1621 . . . . . 6 (z = y → (x(φx = y) → x(φx = z)))
1312com12 27 . . . . 5 (x(φx = y) → (z = yx(φx = z)))
147, 13impbid 183 . . . 4 (x(φx = y) → (x(φx = z) ↔ z = y))
1514alrimiv 1631 . . 3 (x(φx = y) → z(x(φx = z) ↔ z = y))
16 uniabio 4349 . . 3 (z(x(φx = z) ↔ z = y) → {z x(φx = z)} = y)
1715, 16syl 15 . 2 (x(φx = y) → {z x(φx = z)} = y)
181, 17syl5eq 2397 1 (x(φx = y) → (℩xφ) = y)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859  ∪cuni 3891  ℩cio 4337 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  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339 This theorem is referenced by:  iotauni  4351  iota1  4353  iotaex  4356  iota4  4357  iota5  4359
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