Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  setinds2f Structured version   Visualization version   GIF version

Theorem setinds2f 31808
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 3472 . . . . 5 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
2 setinds2f.1 . . . . . 6 𝑥𝜓
3 setinds2f.2 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3sbie 2436 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
51, 4bitr3i 266 . . . 4 ([𝑦 / 𝑥]𝜑𝜓)
65ralbii 3009 . . 3 (∀𝑦𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑥 𝜓)
7 setinds2f.3 . . 3 (∀𝑦𝑥 𝜓𝜑)
86, 7sylbi 207 . 2 (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑)
98setinds 31807 1 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnf 1748  [wsb 1937  wral 2941  [wsbc 3468
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:  setinds2  31809
  Copyright terms: Public domain W3C validator