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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 2174 | . 2 ⊢ (𝜑 → ∃𝑦𝜑) | |
2 | 19.8a 2174 | . 2 ⊢ (∃𝑦𝜑 → ∃𝑥∃𝑦𝜑) | |
3 | nfe1 2147 | . . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑 | |
4 | nfe1 2147 | . . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
5 | 4 | nfex 2317 | . . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦𝜑 |
6 | pgind.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
7 | pgind.2 | . . 3 ⊢ (𝑦 = 𝐴 → (𝜒 ↔ 𝜃)) | |
8 | nfa1 2148 | . . . 4 ⊢ Ⅎ𝑥∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓) | |
9 | nfra1 3281 | . . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 | |
10 | nfv 1917 | . . . . . . 7 ⊢ Ⅎ𝑦𝜓 | |
11 | 9, 10 | nfim 1899 | . . . . . 6 ⊢ Ⅎ𝑦(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓) |
12 | 11 | nfal 2316 | . . . . 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 2210 | . . . 4 ⊢ (∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)) |
15 | 8, 14 | exlimi 2210 | . . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)) |
16 | 3, 5, 6, 7, 15 | pgindnf 47714 | . 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 205 ∀wal 1539 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∀wral 3061 ∪ cun 3945 ‘cfv 6540 1st c1st 7969 2nd c2nd 7970 Pgcpg 47707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2371 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fv 6548 df-1st 7971 df-2nd 7972 df-setrecs 47682 df-pg 47708 |
This theorem is referenced by: (None) |
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