Theorem pgind 51216

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 2221	. 2 ⊢ (𝜑 → ∃𝑦𝜑)
2		19.8a 2221	. 2 ⊢ (∃𝑦𝜑 → ∃𝑥∃𝑦𝜑)
3		nfe1 2190	. . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑
4		nfe1 2190	. . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑
5	4	nfex 2358	. . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦𝜑
6		pgind.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
7		pgind.2	. . 3 ⊢ (𝑦 = 𝐴 → (𝜒 ↔ 𝜃))
8		nfa1 2191	. . . 4 ⊢ Ⅎ𝑥∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)
9		nfra1 3290	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒
10		nfv 1949	. . . . . . 7 ⊢ Ⅎ𝑦𝜓
11	9, 10	nfim 1931	. . . . . 6 ⊢ Ⅎ𝑦(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)
12	11	nfal 2357	. . . . 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 2257	. . . 4 ⊢ (∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))
15	8, 14	exlimi 2257	. . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))
16	3, 5, 6, 7, 15	pgindnf 51215	. 2 ⊢ (∃𝑥∃𝑦𝜑 → (𝐴 ∈ Pg → 𝜃))
17	1, 2, 16	3syl 19	1 ⊢ (𝜑 → (𝐴 ∈ Pg → 𝜃))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080   ∪ cun 4152  ‘cfv 6566  1^st c1st 8033  2^nd c2nd 8034  Pgcpg 51208

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-13 2404  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fv 6574  df-1st 8035  df-2nd 8036  df-setrecs 10002  df-pg 51209

This theorem is used by: (None)