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Theorem fvsb 39127
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 6045 . . 3 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
2 dfsbcq 3566 . . 3 ((𝐹𝐴) = (℩𝑦𝐴𝐹𝑦) → ([(𝐹𝐴) / 𝑥]𝜑[(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑))
31, 2ax-mp 5 . 2 ([(𝐹𝐴) / 𝑥]𝜑[(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑)
4 iotasbc 39091 . 2 (∃!𝑦 𝐴𝐹𝑦 → ([(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
53, 4syl5bb 272 1 (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1618   = wceq 1620  wex 1841  ∃!weu 2595  [wsbc 3564   class class class wbr 4792  cio 5998  cfv 6037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-rex 3044  df-v 3330  df-sbc 3565  df-un 3708  df-sn 4310  df-pr 4312  df-uni 4577  df-iota 6000  df-fv 6045
This theorem is referenced by:  fveqsb  39128
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