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Mirrors > Home > MPE Home > Th. List > wfis3 | Structured version Visualization version GIF version |
Description: Well-Ordered Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.) |
Ref | Expression |
---|---|
wfis3.1 | ⊢ 𝑅 We 𝐴 |
wfis3.2 | ⊢ 𝑅 Se 𝐴 |
wfis3.3 | ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) |
wfis3.4 | ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜒)) |
wfis3.5 | ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) |
Ref | Expression |
---|---|
wfis3 | ⊢ (𝐵 ∈ 𝐴 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfis3.4 | . 2 ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜒)) | |
2 | wfis3.1 | . . 3 ⊢ 𝑅 We 𝐴 | |
3 | wfis3.2 | . . 3 ⊢ 𝑅 Se 𝐴 | |
4 | wfis3.3 | . . 3 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) | |
5 | wfis3.5 | . . 3 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) | |
6 | 2, 3, 4, 5 | wfis2 6358 | . 2 ⊢ (𝑦 ∈ 𝐴 → 𝜑) |
7 | 1, 6 | vtoclga 3565 | 1 ⊢ (𝐵 ∈ 𝐴 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Se wse 5628 We wwe 5629 Predcpred 6296 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-cnv 5683 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 |
This theorem is referenced by: omsinds 7871 omsindsOLD 7872 uzsinds 13948 |
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