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Theorem uniabio 6330
 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 abbi1 2886 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑥𝑥 = 𝑦})
2 df-sn 4570 . . . 4 {𝑦} = {𝑥𝑥 = 𝑦}
31, 2syl6eqr 2876 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
43unieqd 4854 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
5 vex 3499 . . 3 𝑦 ∈ V
65unisn 4860 . 2 {𝑦} = 𝑦
74, 6syl6eq 2874 1 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535   = wceq 1537  {cab 2801  {csn 4569  ∪ cuni 4840 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-un 3943  df-in 3945  df-ss 3954  df-sn 4570  df-pr 4572  df-uni 4841 This theorem is referenced by:  iotaval  6331  iotauni  6332
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