Theorem pgind

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

	Ref	Expression
Hypotheses	pgind.1	⊢ ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) )
	pgind.2	⊢ ( 𝑦 = 𝐴 → ( 𝜒 ↔ 𝜃 ) )
	pgind.3	⊢ ( 𝜑 → ∀ 𝑥 ( ∀ 𝑦 ∈ ( ( 1st ‘ 𝑥 ) ∪ ( 2nd ‘ 𝑥 ) ) 𝜒 → 𝜓 ) )
Assertion	pgind	⊢ ( 𝜑 → ( 𝐴 ∈ Pg → 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		pgind.1	⊢ ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) )
2		pgind.2	⊢ ( 𝑦 = 𝐴 → ( 𝜒 ↔ 𝜃 ) )
3		pgind.3	⊢ ( 𝜑 → ∀ 𝑥 ( ∀ 𝑦 ∈ ( ( 1st ‘ 𝑥 ) ∪ ( 2nd ‘ 𝑥 ) ) 𝜒 → 𝜓 ) )
4		19.8a	⊢ ( 𝜑 → ∃ 𝑦 𝜑 )
5		19.8a	⊢ ( ∃ 𝑦 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜑 )
6		nfe1	⊢ Ⅎ 𝑥 ∃ 𝑥 ∃ 𝑦 𝜑
7		nfe1	⊢ Ⅎ 𝑦 ∃ 𝑦 𝜑
8	7	nfex	⊢ Ⅎ 𝑦 ∃ 𝑥 ∃ 𝑦 𝜑
9		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 ( ∀ 𝑦 ∈ ( ( 1st ‘ 𝑥 ) ∪ ( 2nd ‘ 𝑥 ) ) 𝜒 → 𝜓 )
10		nfra1	⊢ Ⅎ 𝑦 ∀ 𝑦 ∈ ( ( 1st ‘ 𝑥 ) ∪ ( 2nd ‘ 𝑥 ) ) 𝜒
11		nfv	⊢ Ⅎ 𝑦 𝜓
12	10 11	nfim	⊢ Ⅎ 𝑦 ( ∀ 𝑦 ∈ ( ( 1st ‘ 𝑥 ) ∪ ( 2nd ‘ 𝑥 ) ) 𝜒 → 𝜓 )
13	12	nfal	⊢ Ⅎ 𝑦 ∀ 𝑥 ( ∀ 𝑦 ∈ ( ( 1st ‘ 𝑥 ) ∪ ( 2nd ‘ 𝑥 ) ) 𝜒 → 𝜓 )
14	13 3	exlimi	⊢ ( ∃ 𝑦 𝜑 → ∀ 𝑥 ( ∀ 𝑦 ∈ ( ( 1st ‘ 𝑥 ) ∪ ( 2nd ‘ 𝑥 ) ) 𝜒 → 𝜓 ) )
15	9 14	exlimi	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 → ∀ 𝑥 ( ∀ 𝑦 ∈ ( ( 1st ‘ 𝑥 ) ∪ ( 2nd ‘ 𝑥 ) ) 𝜒 → 𝜓 ) )
16	6 8 1 2 15	pgindnf	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 → ( 𝐴 ∈ Pg → 𝜃 ) )
17	4 5 16	3syl	⊢ ( 𝜑 → ( 𝐴 ∈ Pg → 𝜃 ) )