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Theorem scottabes 42614
Description: Value of the Scott operation at a class abstraction. Variant of scottab 42613 using explicit substitution. (Contributed by Rohan Ridenour, 14-Aug-2023.)
Assertion
Ref Expression
scottabes Scott {π‘₯ ∣ πœ‘} = {π‘₯ ∣ (πœ‘ ∧ βˆ€π‘¦([𝑦 / π‘₯]πœ‘ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))}
Distinct variable groups:   πœ‘,𝑦   π‘₯,𝑦
Allowed substitution hint:   πœ‘(π‘₯)

Proof of Theorem scottabes
StepHypRef Expression
1 nfs1v 2154 . 2 β„²π‘₯[𝑦 / π‘₯]πœ‘
2 sbequ12 2244 . 2 (π‘₯ = 𝑦 β†’ (πœ‘ ↔ [𝑦 / π‘₯]πœ‘))
31, 2scottabf 42612 1 Scott {π‘₯ ∣ πœ‘} = {π‘₯ ∣ (πœ‘ ∧ βˆ€π‘¦([𝑦 / π‘₯]πœ‘ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))}
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397  βˆ€wal 1540   = wceq 1542  [wsb 2068  {cab 2710   βŠ† wss 3914  β€˜cfv 6500  rankcrnk 9707  Scott cscott 42607
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-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-scott 42608
This theorem is referenced by: (None)
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