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Theorem uniabio 5764
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 2723 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑦})
21biimpi 204 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑥𝑥 = 𝑦})
3 df-sn 4125 . . . 4 {𝑦} = {𝑥𝑥 = 𝑦}
42, 3syl6eqr 2661 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
54unieqd 4376 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
6 vex 3175 . . 3 𝑦 ∈ V
76unisn 4381 . 2 {𝑦} = 𝑦
85, 7syl6eq 2659 1 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472   = wceq 1474  {cab 2595  {csn 4124   cuni 4366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rex 2901  df-v 3174  df-un 3544  df-sn 4125  df-pr 4127  df-uni 4367
This theorem is referenced by:  iotaval  5765  iotauni  5766
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