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Theorem fveqsb 37678
 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 6113 . . 3 (𝐹𝐴) ∈ V
2 fveqsb.3 . . . 4 𝑥𝜓
3 fveqsb.2 . . . 4 (𝑥 = (𝐹𝐴) → (𝜑𝜓))
42, 3sbciegf 3434 . . 3 ((𝐹𝐴) ∈ V → ([(𝐹𝐴) / 𝑥]𝜑𝜓))
51, 4ax-mp 5 . 2 ([(𝐹𝐴) / 𝑥]𝜑𝜓)
6 fvsb 37677 . 2 (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
75, 6syl5bbr 273 1 (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃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  Ⅎwnf 1699   ∈ wcel 1977  ∃!weu 2458  Vcvv 3173  [wsbc 3402   class class class wbr 4583  ‘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  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  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-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768  df-fv 5812 This theorem is referenced by: (None)
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