Theorem pgind 50779

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 2218	. 2 ⊢ (𝜑 → ∃𝑦𝜑)
2		19.8a 2218	. 2 ⊢ (∃𝑦𝜑 → ∃𝑥∃𝑦𝜑)
3		nfe1 2187	. . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑
4		nfe1 2187	. . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑
5	4	nfex 2355	. . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦𝜑
6		pgind.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
7		pgind.2	. . 3 ⊢ (𝑦 = 𝐴 → (𝜒 ↔ 𝜃))
8		nfa1 2188	. . . 4 ⊢ Ⅎ𝑥∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)
9		nfra1 3287	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒
10		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑦𝜓
11	9, 10	nfim 1929	. . . . . 6 ⊢ Ⅎ𝑦(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓)
12	11	nfal 2354	. . . . 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 2254	. . . 4 ⊢ (∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))
15	8, 14	exlimi 2254	. . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))
16	3, 5, 6, 7, 15	pgindnf 50778	. 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 2145  ∀wral 3077   ∪ cun 3897  ‘cfv 6537  1^st c1st 7997  2^nd c2nd 7998  Pgcpg 50771

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-13 2402  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fv 6545  df-1st 7999  df-2nd 8000  df-setrecs 9958  df-pg 50772

This theorem is used by: (None)