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Theorem uniabio 4905
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 2167 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑦})
21biimpi 117 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑥𝑥 = 𝑦})
3 df-sn 3409 . . . 4 {𝑦} = {𝑥𝑥 = 𝑦}
42, 3syl6eqr 2106 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
54unieqd 3619 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
6 vex 2577 . . 3 𝑦 ∈ V
76unisn 3624 . 2 {𝑦} = 𝑦
85, 7syl6eq 2104 1 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = 𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257   = wceq 1259  {cab 2042  {csn 3403   cuni 3608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-uni 3609
This theorem is referenced by:  iotaval  4906  iotauni  4907
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