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Theorem setinds2f 33098
Description: E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
setinds2f.1 𝑥𝜓
setinds2f.2 (𝑥 = 𝑦 → (𝜑𝜓))
setinds2f.3 (∀𝑦𝑥 𝜓𝜑)
Assertion
Ref Expression
setinds2f 𝜑
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem setinds2f
StepHypRef Expression
1 sbsbc 3751 . . . . 5 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
2 setinds2f.1 . . . . . 6 𝑥𝜓
3 setinds2f.2 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3sbiev 2331 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
51, 4bitr3i 280 . . . 4 ([𝑦 / 𝑥]𝜑𝜓)
65ralbii 3157 . . 3 (∀𝑦𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑥 𝜓)
7 setinds2f.3 . . 3 (∀𝑦𝑥 𝜓𝜑)
86, 7sylbi 220 . 2 (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑)
98setinds 33097 1 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wnf 1785  [wsb 2069  wral 3130  [wsbc 3747
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-reg 9044  ax-inf2 9092
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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033
This theorem is referenced by:  setinds2  33099
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