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

Theorem fvsb 38973
Description: Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
Assertion
Ref Expression
fvsb (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐹,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem fvsb
StepHypRef Expression
1 df-fv 5934 . . 3 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
2 dfsbcq 3470 . . 3 ((𝐹𝐴) = (℩𝑦𝐴𝐹𝑦) → ([(𝐹𝐴) / 𝑥]𝜑[(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑))
31, 2ax-mp 5 . 2 ([(𝐹𝐴) / 𝑥]𝜑[(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑)
4 iotasbc 38937 . 2 (∃!𝑦 𝐴𝐹𝑦 → ([(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
53, 4syl5bb 272 1 (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  ∃!weu 2498  [wsbc 3468   class class class wbr 4685  cio 5887  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233  df-sbc 3469  df-un 3612  df-sn 4211  df-pr 4213  df-uni 4469  df-iota 5889  df-fv 5934
This theorem is referenced by:  fveqsb  38974
  Copyright terms: Public domain W3C validator