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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 6023 | . 2 ⊢ (𝑦 ∈ 𝐴 → 𝜑) |
7 | 1, 6 | vtoclga 3486 | 1 ⊢ (𝐵 ∈ 𝐴 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1508 ∈ wcel 2051 ∀wral 3081 Se wse 5360 We wwe 5361 Predcpred 5982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4926 df-opab 4988 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-cnv 5411 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 |
This theorem is referenced by: omsinds 7413 uzsinds 13168 |
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