Theorem frpoins2g

Description: Founded Partial Induction schema, using implicit substitution. (Contributed by Scott Fenton, 24-Aug-2022)

	Ref	Expression
Hypotheses	frpoins2g.1	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
	frpoins2g.3	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
Assertion	frpoins2g	⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		frpoins2g.1	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
2		frpoins2g.3	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
3		nfv	⊢ Ⅎ 𝑦 𝜓
4	1 3 2	frpoins2fg	⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ 𝐴 𝜑 )