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| Description: Deduction associated with tfinds 7882. (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 1119 | . . . 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 1119 | . . . 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 7887 | . 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 1539 ∈ wcel 2107 ∀wral 3060 ∅c0 4332 Oncon0 6383 Lim wlim 6384 suc csuc 6385 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 | 
| This theorem is referenced by: grur1cld 44256 | 
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