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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 7573 | . 2 ⊢ (𝑥 ∈ On → 𝜑) |
5 | 1, 4 | vtoclga 3576 | 1 ⊢ (𝐴 ∈ On → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Oncon0 6193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-ord 6196 df-on 6197 |
This theorem is referenced by: tfisi 7575 tfinds 7576 tfrlem1 8014 ordtypelem7 8990 rankonidlem 9259 tcrank 9315 infxpenlem 9441 alephle 9516 dfac12lem3 9573 ttukeylem5 9937 ttukeylem6 9938 tskord 10204 grudomon 10241 aomclem6 39666 |
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