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Mirrors > Home > MPE Home > Th. List > Mathboxes > pgind | Structured version Visualization version GIF version |
Description: Induction on partizan games. (Contributed by Emmett Weisz, 27-May-2024.) |
Ref | Expression |
---|---|
pgind.1 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) |
pgind.2 | ⊢ (𝑦 = 𝐴 → (𝜒 ↔ 𝜃)) |
pgind.3 | ⊢ (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)) |
Ref | Expression |
---|---|
pgind | ⊢ (𝜑 → (𝐴 ∈ Pg → 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.8a 2179 | . 2 ⊢ (𝜑 → ∃𝑦𝜑) | |
2 | 19.8a 2179 | . 2 ⊢ (∃𝑦𝜑 → ∃𝑥∃𝑦𝜑) | |
3 | nfe1 2148 | . . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑 | |
4 | nfe1 2148 | . . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
5 | 4 | nfex 2323 | . . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦𝜑 |
6 | pgind.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
7 | pgind.2 | . . 3 ⊢ (𝑦 = 𝐴 → (𝜒 ↔ 𝜃)) | |
8 | nfa1 2149 | . . . 4 ⊢ Ⅎ𝑥∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓) | |
9 | nfra1 3282 | . . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 | |
10 | nfv 1912 | . . . . . . 7 ⊢ Ⅎ𝑦𝜓 | |
11 | 9, 10 | nfim 1894 | . . . . . 6 ⊢ Ⅎ𝑦(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓) |
12 | 11 | nfal 2322 | . . . . 5 ⊢ Ⅎ𝑦∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓) |
13 | pgind.3 | . . . . 5 ⊢ (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)) | |
14 | 12, 13 | exlimi 2215 | . . . 4 ⊢ (∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)) |
15 | 8, 14 | exlimi 2215 | . . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)) |
16 | 3, 5, 6, 7, 15 | pgindnf 48947 | . 2 ⊢ (∃𝑥∃𝑦𝜑 → (𝐴 ∈ Pg → 𝜃)) |
17 | 1, 2, 16 | 3syl 18 | 1 ⊢ (𝜑 → (𝐴 ∈ Pg → 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∀wral 3059 ∪ cun 3961 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 Pgcpg 48940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-13 2375 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-1st 8013 df-2nd 8014 df-setrecs 48915 df-pg 48941 |
This theorem is referenced by: (None) |
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