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| Description: Transfinite Induction with explicit substitution. The first hypothesis is the basis, the second is the induction step for successors, and the third is the induction step for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 5-Mar-2004.) | 
| Ref | Expression | 
|---|---|
| tfindes.1 | ⊢ [∅ / 𝑥]𝜑 | 
| tfindes.2 | ⊢ (𝑥 ∈ On → (𝜑 → [suc 𝑥 / 𝑥]𝜑)) | 
| tfindes.3 | ⊢ (Lim 𝑦 → (∀𝑥 ∈ 𝑦 𝜑 → [𝑦 / 𝑥]𝜑)) | 
| Ref | Expression | 
|---|---|
| tfindes | ⊢ (𝑥 ∈ On → 𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfsbcq 3790 | . 2 ⊢ (𝑦 = ∅ → ([𝑦 / 𝑥]𝜑 ↔ [∅ / 𝑥]𝜑)) | |
| 2 | dfsbcq 3790 | . 2 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
| 3 | dfsbcq 3790 | . 2 ⊢ (𝑦 = suc 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [suc 𝑧 / 𝑥]𝜑)) | |
| 4 | sbceq2a 3800 | . 2 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | |
| 5 | tfindes.1 | . 2 ⊢ [∅ / 𝑥]𝜑 | |
| 6 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ On | |
| 7 | nfsbc1v 3808 | . . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
| 8 | nfsbc1v 3808 | . . . . 5 ⊢ Ⅎ𝑥[suc 𝑧 / 𝑥]𝜑 | |
| 9 | 7, 8 | nfim 1896 | . . . 4 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑) | 
| 10 | 6, 9 | nfim 1896 | . . 3 ⊢ Ⅎ𝑥(𝑧 ∈ On → ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑)) | 
| 11 | eleq1w 2824 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ On ↔ 𝑧 ∈ On)) | |
| 12 | sbceq1a 3799 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
| 13 | suceq 6450 | . . . . . 6 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧) | |
| 14 | 13 | sbceq1d 3793 | . . . . 5 ⊢ (𝑥 = 𝑧 → ([suc 𝑥 / 𝑥]𝜑 ↔ [suc 𝑧 / 𝑥]𝜑)) | 
| 15 | 12, 14 | imbi12d 344 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑))) | 
| 16 | 11, 15 | imbi12d 344 | . . 3 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ On → (𝜑 → [suc 𝑥 / 𝑥]𝜑)) ↔ (𝑧 ∈ On → ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑)))) | 
| 17 | tfindes.2 | . . 3 ⊢ (𝑥 ∈ On → (𝜑 → [suc 𝑥 / 𝑥]𝜑)) | |
| 18 | 10, 16, 17 | chvarfv 2240 | . 2 ⊢ (𝑧 ∈ On → ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑)) | 
| 19 | cbvralsvw 3317 | . . . 4 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑) | |
| 20 | sbsbc 3792 | . . . . 5 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) | |
| 21 | 20 | ralbii 3093 | . . . 4 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑) | 
| 22 | 19, 21 | bitri 275 | . . 3 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑) | 
| 23 | tfindes.3 | . . 3 ⊢ (Lim 𝑦 → (∀𝑥 ∈ 𝑦 𝜑 → [𝑦 / 𝑥]𝜑)) | |
| 24 | 22, 23 | biimtrrid 243 | . 2 ⊢ (Lim 𝑦 → (∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑)) | 
| 25 | 1, 2, 3, 4, 5, 18, 24 | tfinds 7881 | 1 ⊢ (𝑥 ∈ On → 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 [wsb 2064 ∈ wcel 2108 ∀wral 3061 [wsbc 3788 ∅c0 4333 Oncon0 6384 Lim wlim 6385 suc csuc 6386 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 | 
| This theorem is referenced by: tfinds2 7885 rdgssun 37379 | 
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