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Theorem fvsb 42806
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 6509 . . 3 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
2 dfsbcq 3746 . . 3 ((𝐹𝐴) = (℩𝑦𝐴𝐹𝑦) → ([(𝐹𝐴) / 𝑥]𝜑[(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑))
31, 2ax-mp 5 . 2 ([(𝐹𝐴) / 𝑥]𝜑[(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑)
4 iotasbc 42773 . 2 (∃!𝑦 𝐴𝐹𝑦 → ([(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
53, 4bitrid 283 1 (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wex 1782  ∃!weu 2567  [wsbc 3744   class class class wbr 5110  cio 6451  cfv 6501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-sbc 3745  df-un 3920  df-in 3922  df-ss 3932  df-sn 4592  df-pr 4594  df-uni 4871  df-iota 6453  df-fv 6509
This theorem is referenced by:  fveqsb  42807
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