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Mirrors > Home > MPE Home > Th. List > frpoins3g | Structured version Visualization version GIF version |
Description: Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 19-Aug-2024.) |
Ref | Expression |
---|---|
frpoins3g.1 | ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ Pred (𝑅, 𝐴, 𝑥)𝜓 → 𝜑)) |
frpoins3g.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
frpoins3g.3 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) |
Ref | Expression |
---|---|
frpoins3g | ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝐵 ∈ 𝐴) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frpoins3g.1 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ Pred (𝑅, 𝐴, 𝑥)𝜓 → 𝜑)) | |
2 | frpoins3g.2 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | frpoins2g 6340 | . 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑥 ∈ 𝐴 𝜑) |
4 | frpoins3g.3 | . . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
5 | 4 | rspccva 3605 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝐵 ∈ 𝐴) → 𝜒) |
6 | 3, 5 | sylan 579 | 1 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝐵 ∈ 𝐴) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 Po wpo 5579 Fr wfr 5621 Se wse 5622 Predcpred 6293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-po 5581 df-fr 5624 df-se 5625 df-xp 5675 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 |
This theorem is referenced by: noinds 27817 |
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