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Mirrors > Home > MPE Home > Th. List > wfis2f | Structured version Visualization version GIF version |
Description: Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.) |
Ref | Expression |
---|---|
wfis2f.1 | ⊢ 𝑅 We 𝐴 |
wfis2f.2 | ⊢ 𝑅 Se 𝐴 |
wfis2f.3 | ⊢ Ⅎ𝑦𝜓 |
wfis2f.4 | ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) |
wfis2f.5 | ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) |
Ref | Expression |
---|---|
wfis2f | ⊢ (𝑦 ∈ 𝐴 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfis2f.1 | . . 3 ⊢ 𝑅 We 𝐴 | |
2 | wfis2f.2 | . . 3 ⊢ 𝑅 Se 𝐴 | |
3 | wfis2f.3 | . . . 4 ⊢ Ⅎ𝑦𝜓 | |
4 | wfis2f.4 | . . . 4 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) | |
5 | wfis2f.5 | . . . 4 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) | |
6 | 3, 4, 5 | wfis2fg 6187 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) |
7 | 1, 2, 6 | mp2an 690 | . 2 ⊢ ∀𝑦 ∈ 𝐴 𝜑 |
8 | 7 | rspec 3209 | 1 ⊢ (𝑦 ∈ 𝐴 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 Ⅎwnf 1784 ∈ wcel 2114 ∀wral 3140 Se wse 5514 We wwe 5515 Predcpred 6149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 |
This theorem is referenced by: (None) |
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