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Theorem fvsb 41575
 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 6349 . . 3 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
2 dfsbcq 3701 . . 3 ((𝐹𝐴) = (℩𝑦𝐴𝐹𝑦) → ([(𝐹𝐴) / 𝑥]𝜑[(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑))
31, 2ax-mp 5 . 2 ([(𝐹𝐴) / 𝑥]𝜑[(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑)
4 iotasbc 41542 . 2 (∃!𝑦 𝐴𝐹𝑦 → ([(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
53, 4syl5bb 286 1 (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1537   = wceq 1539  ∃wex 1782  ∃!weu 2588  [wsbc 3699   class class class wbr 5037  ℩cio 6298  ‘cfv 6341 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-v 3412  df-sbc 3700  df-un 3866  df-in 3868  df-ss 3878  df-sn 4527  df-pr 4529  df-uni 4803  df-iota 6300  df-fv 6349 This theorem is referenced by:  fveqsb  41576
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