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Mirrors > Home > MPE Home > Th. List > frpoins2fg | Structured version Visualization version GIF version |
Description: Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 24-Aug-2022.) |
Ref | Expression |
---|---|
frpoins2fg.1 | ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) |
frpoins2fg.2 | ⊢ Ⅎ𝑦𝜓 |
frpoins2fg.3 | ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
frpoins2fg | ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbsbc 3721 | . . . . 5 ⊢ ([𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑) | |
2 | frpoins2fg.2 | . . . . . 6 ⊢ Ⅎ𝑦𝜓 | |
3 | frpoins2fg.3 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | sbiev 2309 | . . . . 5 ⊢ ([𝑧 / 𝑦]𝜑 ↔ 𝜓) |
5 | 1, 4 | bitr3i 276 | . . . 4 ⊢ ([𝑧 / 𝑦]𝜑 ↔ 𝜓) |
6 | 5 | ralbii 3092 | . . 3 ⊢ (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓) |
7 | frpoins2fg.1 | . . . 4 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) | |
8 | 7 | adantl 482 | . . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑦 ∈ 𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) |
9 | 6, 8 | syl5bi 241 | . 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑦 ∈ 𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑)) |
10 | 9 | frpoinsg 6241 | 1 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 Ⅎwnf 1786 [wsb 2067 ∈ wcel 2106 ∀wral 3064 [wsbc 3717 Po wpo 5498 Fr wfr 5538 Se wse 5539 Predcpred 6196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3433 df-sbc 3718 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-br 5076 df-opab 5138 df-po 5500 df-fr 5541 df-se 5542 df-xp 5592 df-cnv 5594 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 |
This theorem is referenced by: frpoins2g 6243 wfis2fg 6254 |
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