MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniabio Structured version   Visualization version   GIF version

Theorem uniabio 6022
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 2875 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑦})
21biimpi 206 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑥𝑥 = 𝑦})
3 df-sn 4322 . . . 4 {𝑦} = {𝑥𝑥 = 𝑦}
42, 3syl6eqr 2812 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
54unieqd 4598 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
6 vex 3343 . . 3 𝑦 ∈ V
76unisn 4603 . 2 {𝑦} = 𝑦
85, 7syl6eq 2810 1 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1630   = wceq 1632  {cab 2746  {csn 4321   cuni 4588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-v 3342  df-un 3720  df-sn 4322  df-pr 4324  df-uni 4589
This theorem is referenced by:  iotaval  6023  iotauni  6024
  Copyright terms: Public domain W3C validator