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Mirrors > Home > MPE Home > Th. List > wfis3 | Structured version Visualization version GIF version |
Description: Well Founded 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 6183 | . 2 ⊢ (𝑦 ∈ 𝐴 → 𝜑) |
7 | 1, 6 | vtoclga 3574 | 1 ⊢ (𝐵 ∈ 𝐴 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Se wse 5507 We wwe 5508 Predcpred 6142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-cnv 5558 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 |
This theorem is referenced by: omsinds 7594 uzsinds 13349 |
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