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Theorem fveqsb 43675
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 6904 . . 3 (𝐹𝐴) ∈ V
2 fveqsb.3 . . . 4 𝑥𝜓
3 fveqsb.2 . . . 4 (𝑥 = (𝐹𝐴) → (𝜑𝜓))
42, 3sbciegf 3816 . . 3 ((𝐹𝐴) ∈ V → ([(𝐹𝐴) / 𝑥]𝜑𝜓))
51, 4ax-mp 5 . 2 ([(𝐹𝐴) / 𝑥]𝜑𝜓)
6 fvsb 43674 . 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 205  wa 395  wal 1538   = wceq 1540  wex 1780  wnf 1784  wcel 2105  ∃!weu 2561  Vcvv 3473  [wsbc 3777   class class class wbr 5148  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495  df-fv 6551
This theorem is referenced by: (None)
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