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Theorem fvsb 37677
 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 5812 . . 3 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
2 dfsbcq 3404 . . 3 ((𝐹𝐴) = (℩𝑦𝐴𝐹𝑦) → ([(𝐹𝐴) / 𝑥]𝜑[(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑))
31, 2ax-mp 5 . 2 ([(𝐹𝐴) / 𝑥]𝜑[(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑)
4 iotasbc 37642 . 2 (∃!𝑦 𝐴𝐹𝑦 → ([(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
53, 4syl5bb 271 1 (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695  ∃!weu 2458  [wsbc 3402   class class class wbr 4583  ℩cio 5766  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-sbc 3403  df-un 3545  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768  df-fv 5812 This theorem is referenced by:  fveqsb  37678
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