![]() |
Mathbox for Rohan Ridenour |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > tfindsd | Structured version Visualization version GIF version |
Description: Deduction associated with tfinds 7881. (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 1118 | . . . 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 1118 | . . . 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 7886 | . 2 ⊢ (𝐴 ∈ On → (𝜑 → 𝜂)) |
14 | 1, 13 | mpcom 38 | 1 ⊢ (𝜑 → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∅c0 4339 Oncon0 6386 Lim wlim 6387 suc csuc 6388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 |
This theorem is referenced by: grur1cld 44228 |
Copyright terms: Public domain | W3C validator |