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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 6356 | . 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑥 ∈ 𝐴 𝜑) |
4 | frpoins3g.3 | . . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
5 | 4 | rspccva 3610 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝐵 ∈ 𝐴) → 𝜒) |
6 | 3, 5 | sylan 578 | 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3058 Po wpo 5592 Fr wfr 5634 Se wse 5635 Predcpred 6309 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-po 5594 df-fr 5637 df-se 5638 df-xp 5688 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 |
This theorem is referenced by: noinds 27890 |
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