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

Theorem fveqsb 44363
Description: Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
Hypotheses
Ref Expression
fveqsb.2 (𝑥 = (𝐹𝐴) → (𝜑𝜓))
fveqsb.3 𝑥𝜓
Assertion
Ref Expression
fveqsb (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐹,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem fveqsb
StepHypRef Expression
1 fvex 6932 . . 3 (𝐹𝐴) ∈ V
2 fveqsb.3 . . . 4 𝑥𝜓
3 fveqsb.2 . . . 4 (𝑥 = (𝐹𝐴) → (𝜑𝜓))
42, 3sbciegf 3838 . . 3 ((𝐹𝐴) ∈ V → ([(𝐹𝐴) / 𝑥]𝜑𝜓))
51, 4ax-mp 5 . 2 ([(𝐹𝐴) / 𝑥]𝜑𝜓)
6 fvsb 44362 . 2 (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
75, 6bitr3id 285 1 (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wex 1777  wnf 1781  wcel 2103  ∃!weu 2565  Vcvv 3482  [wsbc 3798   class class class wbr 5169  cfv 6572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-12 2173  ax-ext 2705  ax-nul 5327
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-v 3484  df-sbc 3799  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-sn 4649  df-pr 4651  df-uni 4932  df-iota 6524  df-fv 6580
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator