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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 1830 | . 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 6947 | 1 ⊢ (𝑥 ∈ On → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∈ wcel 1977 ∀wral 2896 Oncon0 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 |
This theorem is referenced by: tfis3 6949 smogt 7351 findcard3 8088 ordiso2 8303 cantnf 8473 cfsmolem 8975 fpwwe2lem8 9338 nqereu 9630 tfis2d 42225 |
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