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Mirrors > Home > MPE Home > Th. List > tfis2 | Structured version Visualization version GIF version |
Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
Ref | Expression |
---|---|
tfis2.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
tfis2.2 | ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)) |
Ref | Expression |
---|---|
tfis2 | ⊢ (𝑥 ∈ On → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1918 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | tfis2.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | tfis2.2 | . 2 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)) | |
4 | 1, 2, 3 | tfis2f 7832 | 1 ⊢ (𝑥 ∈ On → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2107 ∀wral 3062 Oncon0 6356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-tr 5262 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-ord 6359 df-on 6360 |
This theorem is referenced by: tfis3 7834 smogt 8354 findcard3 9273 findcard3OLD 9274 ordiso2 9497 cantnf 9675 cfsmolem 10252 fpwwe2lem7 10619 nqereu 10911 addsprop 27427 negsprop 27476 mulsprop 27553 tfis2d 47565 |
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