New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  uniabio GIF version

Theorem uniabio 4349
 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) → {x φ} = y)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem uniabio
StepHypRef Expression
1 abbi 2463 . . . . 5 (x(φx = y) ↔ {x φ} = {x x = y})
21biimpi 186 . . . 4 (x(φx = y) → {x φ} = {x x = y})
3 df-sn 3741 . . . 4 {y} = {x x = y}
42, 3syl6eqr 2403 . . 3 (x(φx = y) → {x φ} = {y})
54unieqd 3902 . 2 (x(φx = y) → {x φ} = {y})
6 vex 2862 . . 3 y V
76unisn 3907 . 2 {y} = y
85, 7syl6eq 2401 1 (x(φx = y) → {x φ} = y)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642  {cab 2339  {csn 3737  ∪cuni 3891 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892 This theorem is referenced by:  iotaval  4350  iotauni  4351
 Copyright terms: Public domain W3C validator