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Theorem setinds 31807
Description: Principle of E induction (set induction). If a property passes from all elements of 𝑥 to 𝑥 itself, then it holds for all 𝑥. (Contributed by Scott Fenton, 10-Mar-2011.)
Hypothesis
Ref Expression
setinds.1 (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑)
Assertion
Ref Expression
setinds 𝜑
Distinct variable groups:   𝜑,𝑦   𝑥,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem setinds
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 vex 3234 . 2 𝑥 ∈ V
2 setind 8648 . . . . 5 (∀𝑧(𝑧 ⊆ {𝑥𝜑} → 𝑧 ∈ {𝑥𝜑}) → {𝑥𝜑} = V)
3 dfss3 3625 . . . . . . 7 (𝑧 ⊆ {𝑥𝜑} ↔ ∀𝑦𝑧 𝑦 ∈ {𝑥𝜑})
4 df-sbc 3469 . . . . . . . . 9 ([𝑦 / 𝑥]𝜑𝑦 ∈ {𝑥𝜑})
54ralbii 3009 . . . . . . . 8 (∀𝑦𝑧 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑧 𝑦 ∈ {𝑥𝜑})
6 nfcv 2793 . . . . . . . . . . 11 𝑥𝑧
7 nfsbc1v 3488 . . . . . . . . . . 11 𝑥[𝑦 / 𝑥]𝜑
86, 7nfral 2974 . . . . . . . . . 10 𝑥𝑦𝑧 [𝑦 / 𝑥]𝜑
9 nfsbc1v 3488 . . . . . . . . . 10 𝑥[𝑧 / 𝑥]𝜑
108, 9nfim 1865 . . . . . . . . 9 𝑥(∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
11 raleq 3168 . . . . . . . . . 10 (𝑥 = 𝑧 → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑧 [𝑦 / 𝑥]𝜑))
12 sbceq1a 3479 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
1311, 12imbi12d 333 . . . . . . . . 9 (𝑥 = 𝑧 → ((∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ↔ (∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)))
14 setinds.1 . . . . . . . . 9 (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑)
1510, 13, 14chvar 2298 . . . . . . . 8 (∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
165, 15sylbir 225 . . . . . . 7 (∀𝑦𝑧 𝑦 ∈ {𝑥𝜑} → [𝑧 / 𝑥]𝜑)
173, 16sylbi 207 . . . . . 6 (𝑧 ⊆ {𝑥𝜑} → [𝑧 / 𝑥]𝜑)
18 df-sbc 3469 . . . . . 6 ([𝑧 / 𝑥]𝜑𝑧 ∈ {𝑥𝜑})
1917, 18sylib 208 . . . . 5 (𝑧 ⊆ {𝑥𝜑} → 𝑧 ∈ {𝑥𝜑})
202, 19mpg 1764 . . . 4 {𝑥𝜑} = V
2120eqcomi 2660 . . 3 V = {𝑥𝜑}
2221abeq2i 2764 . 2 (𝑥 ∈ V ↔ 𝜑)
231, 22mpbi 220 1 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  {cab 2637  wral 2941  Vcvv 3231  [wsbc 3468  wss 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551
This theorem is referenced by:  setinds2f  31808
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