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Theorem riotasbc 5426
 Description: Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
riotasbc (∃!x A φ[(x A φ) / x]φ)

Proof of Theorem riotasbc
StepHypRef Expression
1 rabssab 3021 . . 3 {x Aφ} ⊆ {xφ}
2 riotacl2 5424 . . 3 (∃!x A φ → (x A φ) {x Aφ})
31, 2sseldi 2937 . 2 (∃!x A φ → (x A φ) {xφ})
4 df-sbc 2759 . 2 ([(x A φ) / x]φ ↔ (x A φ) {xφ})
53, 4sylibr 137 1 (∃!x A φ[(x A φ) / x]φ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1390  {cab 2023  ∃!wreu 2302  {crab 2304  [wsbc 2758  ℩crio 5410 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810  df-riota 5411 This theorem is referenced by:  riotass2  5437  riotass  5438  cjth  9054
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