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Mirrors > Home > MPE Home > Th. List > Mathboxes > tfindsd | Structured version Visualization version GIF version |
Description: Deduction associated with tfinds 7800. (Contributed by Rohan Ridenour, 8-Aug-2023.) |
Ref | Expression |
---|---|
tfindsd.1 | ⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒)) |
tfindsd.2 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) |
tfindsd.3 | ⊢ (𝑥 = suc 𝑦 → (𝜓 ↔ 𝜏)) |
tfindsd.4 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) |
tfindsd.5 | ⊢ (𝜑 → 𝜒) |
tfindsd.6 | ⊢ ((𝜑 ∧ 𝑦 ∈ On ∧ 𝜃) → 𝜏) |
tfindsd.7 | ⊢ ((𝜑 ∧ Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝜃) → 𝜓) |
tfindsd.8 | ⊢ (𝜑 → 𝐴 ∈ On) |
Ref | Expression |
---|---|
tfindsd | ⊢ (𝜑 → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfindsd.8 | . 2 ⊢ (𝜑 → 𝐴 ∈ On) | |
2 | tfindsd.1 | . . 3 ⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒)) | |
3 | tfindsd.2 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) | |
4 | tfindsd.3 | . . 3 ⊢ (𝑥 = suc 𝑦 → (𝜓 ↔ 𝜏)) | |
5 | tfindsd.4 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) | |
6 | tfindsd.5 | . . 3 ⊢ (𝜑 → 𝜒) | |
7 | tfindsd.6 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ On ∧ 𝜃) → 𝜏) | |
8 | 7 | 3exp 1120 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ On → (𝜃 → 𝜏))) |
9 | 8 | com12 32 | . . 3 ⊢ (𝑦 ∈ On → (𝜑 → (𝜃 → 𝜏))) |
10 | tfindsd.7 | . . . . 5 ⊢ ((𝜑 ∧ Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝜃) → 𝜓) | |
11 | 10 | 3exp 1120 | . . . 4 ⊢ (𝜑 → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 𝜃 → 𝜓))) |
12 | 11 | com12 32 | . . 3 ⊢ (Lim 𝑥 → (𝜑 → (∀𝑦 ∈ 𝑥 𝜃 → 𝜓))) |
13 | 2, 3, 4, 5, 6, 9, 12 | tfinds3 7805 | . 2 ⊢ (𝐴 ∈ On → (𝜑 → 𝜂)) |
14 | 1, 13 | mpcom 38 | 1 ⊢ (𝜑 → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∅c0 4286 Oncon0 6321 Lim wlim 6322 suc csuc 6323 |
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 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-tr 5227 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 |
This theorem is referenced by: grur1cld 42604 |
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