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Theorem uniabio 5098
 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 2253 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑦})
21biimpi 119 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑥𝑥 = 𝑦})
3 df-sn 3533 . . . 4 {𝑦} = {𝑥𝑥 = 𝑦}
42, 3syl6eqr 2190 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
54unieqd 3747 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
6 vex 2689 . . 3 𝑦 ∈ V
76unisn 3752 . 2 {𝑦} = 𝑦
85, 7syl6eq 2188 1 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = 𝑦)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329   = wceq 1331  {cab 2125  {csn 3527  ∪ cuni 3736 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-uni 3737 This theorem is referenced by:  iotaval  5099  iotauni  5100
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