Theorem pgind 49703

Description: Induction on partizan games. (Contributed by Emmett Weisz, 27-May-2024.)

Hypotheses

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))

Assertion

Ref	Expression
pgind	⊢ (𝜑 → (𝐴 ∈ Pg → 𝜃))

Distinct variable groups:   𝑦,𝐴   𝜒,𝑥   𝜓,𝑦   𝜃,𝑦   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑥)   𝐴(𝑥)

Proof of Theorem pgind

Step	Hyp	Ref	Expression
1		19.8a 2182	. 2 ⊢ (𝜑 → ∃𝑦𝜑)
2		19.8a 2182	. 2 ⊢ (∃𝑦𝜑 → ∃𝑥∃𝑦𝜑)
3		nfe1 2151	. . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑
4		nfe1 2151	. . . 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 2152	. . . 4 ⊢ Ⅎ𝑥∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)
9		nfra1 3253	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒
10		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑦𝜓
11	9, 10	nfim 1896	. . . . . 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 2218	. . . 4 ⊢ (∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))
15	8, 14	exlimi 2218	. . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))
16	3, 5, 6, 7, 15	pgindnf 49702	. 2 ⊢ (∃𝑥∃𝑦𝜑 → (𝐴 ∈ Pg → 𝜃))
17	1, 2, 16	3syl 18	1 ⊢ (𝜑 → (𝐴 ∈ Pg → 𝜃))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044   ∪ cun 3903  ‘cfv 6486  1^st c1st 7929  2^nd c2nd 7930  Pgcpg 49695

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2370  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fv 6494  df-1st 7931  df-2nd 7932  df-setrecs 49670  df-pg 49696

This theorem is referenced by: (None)