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Mirrors > Home > MPE Home > Th. List > wfis2 | Structured version Visualization version GIF version |
Description: Well-Ordered Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.) |
Ref | Expression |
---|---|
wfis2.1 | ⊢ 𝑅 We 𝐴 |
wfis2.2 | ⊢ 𝑅 Se 𝐴 |
wfis2.3 | ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) |
wfis2.4 | ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) |
Ref | Expression |
---|---|
wfis2 | ⊢ (𝑦 ∈ 𝐴 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfis2.1 | . . 3 ⊢ 𝑅 We 𝐴 | |
2 | wfis2.2 | . . 3 ⊢ 𝑅 Se 𝐴 | |
3 | wfis2.3 | . . . 4 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) | |
4 | wfis2.4 | . . . 4 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) | |
5 | 3, 4 | wfis2g 6262 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) |
6 | 1, 2, 5 | mp2an 689 | . 2 ⊢ ∀𝑦 ∈ 𝐴 𝜑 |
7 | 6 | rspec 3133 | 1 ⊢ (𝑦 ∈ 𝐴 → 𝜑) |
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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