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| Mirrors > Home > MPE Home > Th. List > tfis3 | Structured version Visualization version GIF version | ||
| Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.) |
| Ref | Expression |
|---|---|
| tfis3.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
| tfis3.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) |
| tfis3.3 | ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)) |
| Ref | Expression |
|---|---|
| tfis3 | ⊢ (𝐴 ∈ On → 𝜒) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tfis3.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
| 2 | tfis3.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 3 | tfis3.3 | . . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)) | |
| 4 | 2, 3 | tfis2 7878 | . 2 ⊢ (𝑥 ∈ On → 𝜑) |
| 5 | 1, 4 | vtoclga 3577 | 1 ⊢ (𝐴 ∈ On → 𝜒) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Oncon0 6384 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: tfisi 7880 tfinds 7881 tfrlem1 8416 naddrid 8721 naddssim 8723 ordtypelem7 9564 rankonidlem 9868 tcrank 9924 infxpenlem 10053 alephle 10128 dfac12lem3 10186 ttukeylem5 10553 ttukeylem6 10554 tskord 10820 grudomon 10857 madebdayim 27926 madebday 27938 aomclem6 43071 nadd1suc 43405 |
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