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Mirrors > Home > MPE Home > Th. List > Mathboxes > tfis2d | Structured version Visualization version GIF version |
Description: Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.) |
Ref | Expression |
---|---|
tfis2d.1 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
tfis2d.2 | ⊢ (𝜑 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜒 → 𝜓))) |
Ref | Expression |
---|---|
tfis2d | ⊢ (𝜑 → (𝑥 ∈ On → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfis2d.1 | . . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
2 | 1 | com12 32 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 ↔ 𝜒))) |
3 | 2 | pm5.74d 272 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
4 | r19.21v 3174 | . . . 4 ⊢ (∀𝑦 ∈ 𝑥 (𝜑 → 𝜒) ↔ (𝜑 → ∀𝑦 ∈ 𝑥 𝜒)) | |
5 | tfis2d.2 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜒 → 𝜓))) | |
6 | 5 | com12 32 | . . . . 5 ⊢ (𝑥 ∈ On → (𝜑 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜓))) |
7 | 6 | a2d 29 | . . . 4 ⊢ (𝑥 ∈ On → ((𝜑 → ∀𝑦 ∈ 𝑥 𝜒) → (𝜑 → 𝜓))) |
8 | 4, 7 | biimtrid 241 | . . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝜑 → 𝜒) → (𝜑 → 𝜓))) |
9 | 3, 8 | tfis2 7785 | . 2 ⊢ (𝑥 ∈ On → (𝜑 → 𝜓)) |
10 | 9 | com12 32 | 1 ⊢ (𝜑 → (𝑥 ∈ On → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 ∀wral 3062 Oncon0 6315 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-tr 5221 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-ord 6318 df-on 6319 |
This theorem is referenced by: (None) |
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