![]() |
Mathbox for Emmett Weisz |
< Previous
Next >
Nearby theorems |
|
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 2170 | . 2 ⊢ (𝜑 → ∃𝑦𝜑) | |
2 | 19.8a 2170 | . 2 ⊢ (∃𝑦𝜑 → ∃𝑥∃𝑦𝜑) | |
3 | nfe1 2140 | . . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑 | |
4 | nfe1 2140 | . . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
5 | 4 | nfex 2313 | . . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦𝜑 |
6 | pgind.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
7 | pgind.2 | . . 3 ⊢ (𝑦 = 𝐴 → (𝜒 ↔ 𝜃)) | |
8 | nfa1 2141 | . . . 4 ⊢ Ⅎ𝑥∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓) | |
9 | nfra1 3277 | . . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 | |
10 | nfv 1910 | . . . . . . 7 ⊢ Ⅎ𝑦𝜓 | |
11 | 9, 10 | nfim 1892 | . . . . . 6 ⊢ Ⅎ𝑦(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓) |
12 | 11 | nfal 2312 | . . . . 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 2206 | . . . 4 ⊢ (∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)) |
15 | 8, 14 | exlimi 2206 | . . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)) |
16 | 3, 5, 6, 7, 15 | pgindnf 48141 | . 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 1532 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ∀wral 3057 ∪ cun 3943 ‘cfv 6542 1st c1st 7985 2nd c2nd 7986 Pgcpg 48134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-13 2367 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-1st 7987 df-2nd 7988 df-setrecs 48109 df-pg 48135 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |