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Theorem wfis2 6263
Description: Well-Ordered Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
Hypotheses
Ref Expression
wfis2.1 𝑅 We 𝐴
wfis2.2 𝑅 Se 𝐴
wfis2.3 (𝑦 = 𝑧 → (𝜑𝜓))
wfis2.4 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
Assertion
Ref Expression
wfis2 (𝑦𝐴𝜑)
Distinct variable groups:   𝑦,𝐴,𝑧   𝜑,𝑧   𝜓,𝑦   𝑦,𝑅,𝑧
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑧)

Proof of Theorem wfis2
StepHypRef Expression
1 wfis2.1 . . 3 𝑅 We 𝐴
2 wfis2.2 . . 3 𝑅 Se 𝐴
3 wfis2.3 . . . 4 (𝑦 = 𝑧 → (𝜑𝜓))
4 wfis2.4 . . . 4 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
53, 4wfis2g 6262 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
61, 2, 5mp2an 689 . 2 𝑦𝐴 𝜑
76rspec 3133 1 (𝑦𝐴𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2106  wral 3064   Se wse 5542   We wwe 5543  Predcpred 6201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202
This theorem is referenced by:  wfis3  6264
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