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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 3744 | . . . . 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 277 | . . . 4 ⊢ ([𝑧 / 𝑦]𝜑 ↔ 𝜓) |
6 | 5 | ralbii 3093 | . . 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 483 | . . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑦 ∈ 𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) |
9 | 6, 8 | biimtrid 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 6298 | 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 397 ∧ w3a 1088 Ⅎwnf 1786 [wsb 2068 ∈ wcel 2107 ∀wral 3061 [wsbc 3740 Po wpo 5544 Fr wfr 5586 Se wse 5587 Predcpred 6253 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-po 5546 df-fr 5589 df-se 5590 df-xp 5640 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 |
This theorem is referenced by: frpoins2g 6300 wfis2fg 6311 |
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