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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 275 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
4 | r19.21v 3175 | . . . 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 | syl5bi 244 | . . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝜑 → 𝜒) → (𝜑 → 𝜓))) |
9 | 3, 8 | tfis2 7571 | . 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 208 ∈ wcel 2114 ∀wral 3138 Oncon0 6191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-ord 6194 df-on 6195 |
This theorem is referenced by: (None) |
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