Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iotavalsb Structured version   Visualization version   GIF version

Theorem iotavalsb 44457
Description: Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotavalsb (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem iotavalsb
StepHypRef Expression
1 19.8a 2180 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 eu6 2573 . . 3 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
3 iotavalb 44454 . . . 4 (∃!𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
4 dfsbcq 3789 . . . . 5 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓))
54eqcoms 2744 . . . 4 ((℩𝑥𝜑) = 𝑦 → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓))
63, 5biimtrdi 253 . . 3 (∃!𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓)))
72, 6sylbir 235 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓)))
81, 7mpcom 38 1 (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓[(℩𝑥𝜑) / 𝑧]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1537   = wceq 1539  wex 1778  ∃!weu 2567  [wsbc 3787  cio 6511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-sbc 3788  df-un 3955  df-ss 3967  df-sn 4626  df-pr 4628  df-uni 4907  df-iota 6513
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator