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Mirrors > Home > MPE Home > Th. List > Mathboxes > tfindsd | Structured version Visualization version GIF version |
Description: Deduction associated with tfinds 7554. (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 1116 | . . . 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 1116 | . . . 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 7559 | . 2 ⊢ (𝐴 ∈ On → (𝜑 → 𝜂)) |
14 | 1, 13 | mpcom 38 | 1 ⊢ (𝜑 → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∅c0 4243 Oncon0 6159 Lim wlim 6160 suc csuc 6161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 |
This theorem is referenced by: grur1cld 40940 |
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