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Theorem setinds 33050
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 3483 . 2 𝑥 ∈ V
2 setind 9169 . . . . 5 (∀𝑧(𝑧 ⊆ {𝑥𝜑} → 𝑧 ∈ {𝑥𝜑}) → {𝑥𝜑} = V)
3 dfss3 3941 . . . . . . 7 (𝑧 ⊆ {𝑥𝜑} ↔ ∀𝑦𝑧 𝑦 ∈ {𝑥𝜑})
4 df-sbc 3759 . . . . . . . . 9 ([𝑦 / 𝑥]𝜑𝑦 ∈ {𝑥𝜑})
54ralbii 3160 . . . . . . . 8 (∀𝑦𝑧 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑧 𝑦 ∈ {𝑥𝜑})
6 nfcv 2982 . . . . . . . . . . 11 𝑥𝑧
7 nfsbc1v 3778 . . . . . . . . . . 11 𝑥[𝑦 / 𝑥]𝜑
86, 7nfralw 3220 . . . . . . . . . 10 𝑥𝑦𝑧 [𝑦 / 𝑥]𝜑
9 nfsbc1v 3778 . . . . . . . . . 10 𝑥[𝑧 / 𝑥]𝜑
108, 9nfim 1898 . . . . . . . . 9 𝑥(∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
11 raleq 3397 . . . . . . . . . 10 (𝑥 = 𝑧 → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑧 [𝑦 / 𝑥]𝜑))
12 sbceq1a 3769 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
1311, 12imbi12d 348 . . . . . . . . 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 2244 . . . . . . . 8 (∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
165, 15sylbir 238 . . . . . . 7 (∀𝑦𝑧 𝑦 ∈ {𝑥𝜑} → [𝑧 / 𝑥]𝜑)
173, 16sylbi 220 . . . . . 6 (𝑧 ⊆ {𝑥𝜑} → [𝑧 / 𝑥]𝜑)
18 df-sbc 3759 . . . . . 6 ([𝑧 / 𝑥]𝜑𝑧 ∈ {𝑥𝜑})
1917, 18sylib 221 . . . . 5 (𝑧 ⊆ {𝑥𝜑} → 𝑧 ∈ {𝑥𝜑})
202, 19mpg 1799 . . . 4 {𝑥𝜑} = V
2120eqcomi 2833 . . 3 V = {𝑥𝜑}
2221abeq2i 2951 . 2 (𝑥 ∈ V ↔ 𝜑)
231, 22mpbi 233 1 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  {cab 2802  wral 3133  Vcvv 3480  [wsbc 3758  wss 3919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-reg 9049  ax-inf2 9097
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-om 7572  df-wrecs 7939  df-recs 8000  df-rdg 8038
This theorem is referenced by:  setinds2f  33051
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