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Theorem wfis2fg 5902
Description: Well-Founded Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.)
Hypotheses
Ref Expression
wfis2fg.1 𝑦𝜓
wfis2fg.2 (𝑦 = 𝑧 → (𝜑𝜓))
wfis2fg.3 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
Assertion
Ref Expression
wfis2fg ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Distinct variable groups:   𝑦,𝐴,𝑧   𝜑,𝑧   𝑦,𝑅,𝑧
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑦,𝑧)

Proof of Theorem wfis2fg
StepHypRef Expression
1 sbsbc 3600 . . . . 5 ([𝑧 / 𝑦]𝜑[𝑧 / 𝑦]𝜑)
2 wfis2fg.1 . . . . . 6 𝑦𝜓
3 wfis2fg.2 . . . . . 6 (𝑦 = 𝑧 → (𝜑𝜓))
42, 3sbie 2499 . . . . 5 ([𝑧 / 𝑦]𝜑𝜓)
51, 4bitr3i 268 . . . 4 ([𝑧 / 𝑦]𝜑𝜓)
65ralbii 3127 . . 3 (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓)
7 wfis2fg.3 . . 3 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
86, 7syl5bi 233 . 2 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
98wfisg 5900 1 ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wnf 1878  [wsb 2062  wcel 2155  wral 3055  [wsbc 3596   Se wse 5234   We wwe 5235  Predcpred 5864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865
This theorem is referenced by:  wfis2f  5903  wfis2g  5904
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