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Theorem uniabio 4820
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 2148 . . . . 5 (x(φx = y) ↔ {xφ} = {xx = y})
21biimpi 113 . . . 4 (x(φx = y) → {xφ} = {xx = y})
3 df-sn 3373 . . . 4 {y} = {xx = y}
42, 3syl6eqr 2087 . . 3 (x(φx = y) → {xφ} = {y})
54unieqd 3582 . 2 (x(φx = y) → {xφ} = {y})
6 vex 2554 . . 3 y V
76unisn 3587 . 2 {y} = y
85, 7syl6eq 2085 1 (x(φx = y) → {xφ} = y)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242  {cab 2023  {csn 3367   cuni 3571
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-bnd 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-un 2916  df-sn 3373  df-pr 3374  df-uni 3572
This theorem is referenced by:  iotaval  4821  iotauni  4822
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