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Theorem iotaval 4821
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 4811 . 2 (℩xφ) = {zx(φx = z)}
2 vex 2554 . . . . . . 7 y V
3 sbeqalb 2809 . . . . . . . 8 (y V → ((x(φx = y) x(φx = z)) → y = z))
4 equcomi 1589 . . . . . . . 8 (y = zz = y)
53, 4syl6 29 . . . . . . 7 (y V → ((x(φx = y) x(φx = z)) → z = y))
62, 5ax-mp 7 . . . . . 6 ((x(φx = y) x(φx = z)) → z = y)
76ex 108 . . . . 5 (x(φx = y) → (x(φx = z) → z = y))
8 equequ2 1596 . . . . . . . . . 10 (y = z → (x = yx = z))
98equcoms 1591 . . . . . . . . 9 (z = y → (x = yx = z))
109bibi2d 221 . . . . . . . 8 (z = y → ((φx = y) ↔ (φx = z)))
1110biimpd 132 . . . . . . 7 (z = y → ((φx = y) → (φx = z)))
1211alimdv 1756 . . . . . 6 (z = y → (x(φx = y) → x(φx = z)))
1312com12 27 . . . . 5 (x(φx = y) → (z = yx(φx = z)))
147, 13impbid 120 . . . 4 (x(φx = y) → (x(φx = z) ↔ z = y))
1514alrimiv 1751 . . 3 (x(φx = y) → z(x(φx = z) ↔ z = y))
16 uniabio 4820 . . 3 (z(x(φx = z) ↔ z = y) → {zx(φx = z)} = y)
1715, 16syl 14 . 2 (x(φx = y) → {zx(φx = z)} = y)
181, 17syl5eq 2081 1 (x(φx = y) → (℩xφ) = y)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242   wcel 1390  {cab 2023  Vcvv 2551   cuni 3571  cio 4808
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810
This theorem is referenced by:  iotauni  4822  iota1  4824  euiotaex  4826  iota4  4828  iota5  4830
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