Theorem pgind 50221

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 2195	. 2 ⊢ (𝜑 → ∃𝑦𝜑)
2		19.8a 2195	. 2 ⊢ (∃𝑦𝜑 → ∃𝑥∃𝑦𝜑)
3		nfe1 2163	. . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑
4		nfe1 2163	. . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑
5	4	nfex 2335	. . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦𝜑
6		pgind.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
7		pgind.2	. . 3 ⊢ (𝑦 = 𝐴 → (𝜒 ↔ 𝜃))
8		nfa1 2164	. . . 4 ⊢ Ⅎ𝑥∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)
9		nfra1 3265	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒
10		nfv 1922	. . . . . . 7 ⊢ Ⅎ𝑦𝜓
11	9, 10	nfim 1904	. . . . . 6 ⊢ Ⅎ𝑦(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)
12	11	nfal 2334	. . . . 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 2231	. . . 4 ⊢ (∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))
15	8, 14	exlimi 2231	. . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))
16	3, 5, 6, 7, 15	pgindnf 50220	. 2 ⊢ (∃𝑥∃𝑦𝜑 → (𝐴 ∈ Pg → 𝜃))
17	1, 2, 16	3syl 18	1 ⊢ (𝜑 → (𝐴 ∈ Pg → 𝜃))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1546   = wceq 1548  ∃wex 1787   ∈ wcel 2121  ∀wral 3055   ∪ cun 3883  ‘cfv 6489  1^st c1st 7933  2^nd c2nd 7934  Pgcpg 50213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-13 2382  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  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 6445  df-fun 6491  df-fv 6497  df-1st 7935  df-2nd 7936  df-setrecs 50188  df-pg 50214

This theorem is referenced by: (None)