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Theorem uniabio 4804
Description: Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
uniabio (x(φx = y) → {xφ} = y)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem uniabio
StepHypRef Expression
1 abbi 2133 . . . . 5 (x(φx = y) ↔ {xφ} = {xx = y})
21biimpi 113 . . . 4 (x(φx = y) → {xφ} = {xx = y})
3 df-sn 3356 . . . 4 {y} = {xx = y}
42, 3syl6eqr 2072 . . 3 (x(φx = y) → {xφ} = {y})
54unieqd 3565 . 2 (x(φx = y) → {xφ} = {y})
6 vex 2538 . . 3 y V
76unisn 3570 . 2 {y} = y
85, 7syl6eq 2070 1 (x(φx = y) → {xφ} = y)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1226   = wceq 1228  {cab 2008  {csn 3350   cuni 3554
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357  df-uni 3555
This theorem is referenced by:  iotaval  4805  iotauni  4806
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