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Theorem setindel 4423
Description: -Induction in terms of membership in a class. (Contributed by Mario Carneiro and Jim Kingdon, 22-Oct-2018.)
Assertion
Ref Expression
setindel (∀𝑥(∀𝑦(𝑦𝑥𝑦𝑆) → 𝑥𝑆) → 𝑆 = V)
Distinct variable group:   𝑥,𝑦,𝑆

Proof of Theorem setindel
StepHypRef Expression
1 clelsb3 2222 . . . . . . 7 ([𝑦 / 𝑥]𝑥𝑆𝑦𝑆)
21ralbii 2418 . . . . . 6 (∀𝑦𝑥 [𝑦 / 𝑥]𝑥𝑆 ↔ ∀𝑦𝑥 𝑦𝑆)
3 df-ral 2398 . . . . . 6 (∀𝑦𝑥 𝑦𝑆 ↔ ∀𝑦(𝑦𝑥𝑦𝑆))
42, 3bitri 183 . . . . 5 (∀𝑦𝑥 [𝑦 / 𝑥]𝑥𝑆 ↔ ∀𝑦(𝑦𝑥𝑦𝑆))
54imbi1i 237 . . . 4 ((∀𝑦𝑥 [𝑦 / 𝑥]𝑥𝑆𝑥𝑆) ↔ (∀𝑦(𝑦𝑥𝑦𝑆) → 𝑥𝑆))
65albii 1431 . . 3 (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝑥𝑆𝑥𝑆) ↔ ∀𝑥(∀𝑦(𝑦𝑥𝑦𝑆) → 𝑥𝑆))
7 ax-setind 4422 . . 3 (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝑥𝑆𝑥𝑆) → ∀𝑥 𝑥𝑆)
86, 7sylbir 134 . 2 (∀𝑥(∀𝑦(𝑦𝑥𝑦𝑆) → 𝑥𝑆) → ∀𝑥 𝑥𝑆)
9 eqv 3352 . 2 (𝑆 = V ↔ ∀𝑥 𝑥𝑆)
108, 9sylibr 133 1 (∀𝑥(∀𝑦(𝑦𝑥𝑦𝑆) → 𝑥𝑆) → 𝑆 = V)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1314   = wceq 1316  wcel 1465  [wsb 1720  wral 2393  Vcvv 2660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-ral 2398  df-v 2662
This theorem is referenced by:  setind  4424
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