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Theorem wfis2fg 5860
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 3591 . . . . 5 ([𝑧 / 𝑦]𝜑[𝑧 / 𝑦]𝜑)
2 wfis2fg.1 . . . . . 6 𝑦𝜓
3 wfis2fg.2 . . . . . 6 (𝑦 = 𝑧 → (𝜑𝜓))
42, 3sbie 2555 . . . . 5 ([𝑧 / 𝑦]𝜑𝜓)
51, 4bitr3i 266 . . . 4 ([𝑧 / 𝑦]𝜑𝜓)
65ralbii 3129 . . 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 232 . 2 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
98wfisg 5858 1 ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wnf 1856  [wsb 2049  wcel 2145  wral 3061  [wsbc 3587   Se wse 5206   We wwe 5207  Predcpred 5822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823
This theorem is referenced by:  wfis2f  5861  wfis2g  5862
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