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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 7877 | . 2 ⊢ (𝑥 ∈ On → 𝜑) |
5 | 1, 4 | vtoclga 3576 | 1 ⊢ (𝐴 ∈ On → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 ∀wral 3058 Oncon0 6385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-ord 6388 df-on 6389 |
This theorem is referenced by: tfisi 7879 tfinds 7880 tfrlem1 8414 naddrid 8719 naddssim 8721 ordtypelem7 9561 rankonidlem 9865 tcrank 9921 infxpenlem 10050 alephle 10125 dfac12lem3 10183 ttukeylem5 10550 ttukeylem6 10551 tskord 10817 grudomon 10854 madebdayim 27940 madebday 27952 aomclem6 43047 nadd1suc 43381 |
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