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Theorem setinds 32920
Description: Principle of set induction (or E-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 3495 . 2 𝑥 ∈ V
2 setind 9164 . . . . 5 (∀𝑧(𝑧 ⊆ {𝑥𝜑} → 𝑧 ∈ {𝑥𝜑}) → {𝑥𝜑} = V)
3 dfss3 3953 . . . . . . 7 (𝑧 ⊆ {𝑥𝜑} ↔ ∀𝑦𝑧 𝑦 ∈ {𝑥𝜑})
4 df-sbc 3770 . . . . . . . . 9 ([𝑦 / 𝑥]𝜑𝑦 ∈ {𝑥𝜑})
54ralbii 3162 . . . . . . . 8 (∀𝑦𝑧 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑧 𝑦 ∈ {𝑥𝜑})
6 nfcv 2974 . . . . . . . . . . 11 𝑥𝑧
7 nfsbc1v 3789 . . . . . . . . . . 11 𝑥[𝑦 / 𝑥]𝜑
86, 7nfralw 3222 . . . . . . . . . 10 𝑥𝑦𝑧 [𝑦 / 𝑥]𝜑
9 nfsbc1v 3789 . . . . . . . . . 10 𝑥[𝑧 / 𝑥]𝜑
108, 9nfim 1888 . . . . . . . . 9 𝑥(∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
11 raleq 3403 . . . . . . . . . 10 (𝑥 = 𝑧 → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑧 [𝑦 / 𝑥]𝜑))
12 sbceq1a 3780 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
1311, 12imbi12d 346 . . . . . . . . 9 (𝑥 = 𝑧 → ((∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ↔ (∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)))
14 setinds.1 . . . . . . . . 9 (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑)
1510, 13, 14chvarfv 2232 . . . . . . . 8 (∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
165, 15sylbir 236 . . . . . . 7 (∀𝑦𝑧 𝑦 ∈ {𝑥𝜑} → [𝑧 / 𝑥]𝜑)
173, 16sylbi 218 . . . . . 6 (𝑧 ⊆ {𝑥𝜑} → [𝑧 / 𝑥]𝜑)
18 df-sbc 3770 . . . . . 6 ([𝑧 / 𝑥]𝜑𝑧 ∈ {𝑥𝜑})
1917, 18sylib 219 . . . . 5 (𝑧 ⊆ {𝑥𝜑} → 𝑧 ∈ {𝑥𝜑})
202, 19mpg 1789 . . . 4 {𝑥𝜑} = V
2120eqcomi 2827 . . 3 V = {𝑥𝜑}
2221abeq2i 2945 . 2 (𝑥 ∈ V ↔ 𝜑)
231, 22mpbi 231 1 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  {cab 2796  wral 3135  Vcvv 3492  [wsbc 3769  wss 3933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035
This theorem is referenced by:  setinds2f  32921
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