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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 5717 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) |
7 | 1, 2, 6 | mp2an 708 | . 2 ⊢ ∀𝑦 ∈ 𝐴 𝜑 |
8 | 7 | rspec 2931 | 1 ⊢ (𝑦 ∈ 𝐴 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 Ⅎwnf 1708 ∈ wcel 1990 ∀wral 2912 Se wse 5071 We wwe 5072 Predcpred 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 |
This theorem is referenced by: (None) |
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