Theorem pgind 49845

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 2186	. 2 ⊢ (𝜑 → ∃𝑦𝜑)
2		19.8a 2186	. 2 ⊢ (∃𝑦𝜑 → ∃𝑥∃𝑦𝜑)
3		nfe1 2155	. . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑
4		nfe1 2155	. . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑
5	4	nfex 2327	. . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦𝜑
6		pgind.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
7		pgind.2	. . 3 ⊢ (𝑦 = 𝐴 → (𝜒 ↔ 𝜃))
8		nfa1 2156	. . . 4 ⊢ Ⅎ𝑥∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)
9		nfra1 3257	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒
10		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑦𝜓
11	9, 10	nfim 1897	. . . . . 6 ⊢ Ⅎ𝑦(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)
12	11	nfal 2326	. . . . 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 2222	. . . 4 ⊢ (∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))
15	8, 14	exlimi 2222	. . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))
16	3, 5, 6, 7, 15	pgindnf 49844	. 2 ⊢ (∃𝑥∃𝑦𝜑 → (𝐴 ∈ Pg → 𝜃))
17	1, 2, 16	3syl 18	1 ⊢ (𝜑 → (𝐴 ∈ Pg → 𝜃))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3048   ∪ cun 3896  ‘cfv 6488  1^st c1st 7927  2^nd c2nd 7928  Pgcpg 49837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-13 2374  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6444  df-fun 6490  df-fv 6496  df-1st 7929  df-2nd 7930  df-setrecs 49812  df-pg 49838

This theorem is referenced by: (None)