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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 6060 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) |
7 | 1, 2, 6 | mp2an 688 | . 2 ⊢ ∀𝑦 ∈ 𝐴 𝜑 |
8 | 7 | rspec 3174 | 1 ⊢ (𝑦 ∈ 𝐴 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 Ⅎwnf 1765 ∈ wcel 2081 ∀wral 3105 Se wse 5400 We wwe 5401 Predcpred 6022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-br 4963 df-opab 5025 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-cnv 5451 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 |
This theorem is referenced by: (None) |
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