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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 7613 | . 2 ⊢ (𝑥 ∈ On → 𝜑) |
5 | 1, 4 | vtoclga 3479 | 1 ⊢ (𝐴 ∈ On → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 ∀wral 3051 Oncon0 6191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-tr 5147 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-ord 6194 df-on 6195 |
This theorem is referenced by: tfisi 7615 tfinds 7616 tfrlem1 8090 ordtypelem7 9118 rankonidlem 9409 tcrank 9465 infxpenlem 9592 alephle 9667 dfac12lem3 9724 ttukeylem5 10092 ttukeylem6 10093 tskord 10359 grudomon 10396 naddid1 33522 naddssim 33523 madebdayim 33756 madebday 33766 aomclem6 40528 |
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