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 7567. (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 1114 | . . . 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 1114 | . . . 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 7572 | . 2 ⊢ (𝐴 ∈ On → (𝜑 → 𝜂)) |
14 | 1, 13 | mpcom 38 | 1 ⊢ (𝜑 → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ∅c0 4284 Oncon0 6184 Lim wlim 6185 suc csuc 6186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-tr 5166 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 |
This theorem is referenced by: grur1cld 40642 |
Copyright terms: Public domain | W3C validator |