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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 2166 | . 2 ⊢ (𝜑 → ∃𝑦𝜑) | |
2 | 19.8a 2166 | . 2 ⊢ (∃𝑦𝜑 → ∃𝑥∃𝑦𝜑) | |
3 | nfe1 2139 | . . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑 | |
4 | nfe1 2139 | . . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
5 | 4 | nfex 2309 | . . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦𝜑 |
6 | pgind.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
7 | pgind.2 | . . 3 ⊢ (𝑦 = 𝐴 → (𝜒 ↔ 𝜃)) | |
8 | nfa1 2140 | . . . 4 ⊢ Ⅎ𝑥∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓) | |
9 | nfra1 3273 | . . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 | |
10 | nfv 1909 | . . . . . . 7 ⊢ Ⅎ𝑦𝜓 | |
11 | 9, 10 | nfim 1891 | . . . . . 6 ⊢ Ⅎ𝑦(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓) |
12 | 11 | nfal 2308 | . . . . 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 2202 | . . . 4 ⊢ (∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)) |
15 | 8, 14 | exlimi 2202 | . . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)) |
16 | 3, 5, 6, 7, 15 | pgindnf 48008 | . 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 1531 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∀wral 3053 ∪ cun 3939 ‘cfv 6534 1st c1st 7967 2nd c2nd 7968 Pgcpg 48001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-13 2363 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fv 6542 df-1st 7969 df-2nd 7970 df-setrecs 47976 df-pg 48002 |
This theorem is referenced by: (None) |
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