Theorem frins

Description: Founded Induction Schema. If a property passes from all elements less than y of a founded class A to y itself (induction hypothesis), then the property holds for all elements of A . (Contributed by Scott Fenton, 6-Feb-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	frins.1	⊢ 𝑅 Fr 𝐴
	frins.2	⊢ 𝑅 Se 𝐴
	frins.3	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) [ 𝑧 / 𝑦 ] 𝜑 → 𝜑 ) )
Assertion	frins	⊢ ( 𝑦 ∈ 𝐴 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		frins.1	⊢ 𝑅 Fr 𝐴
2		frins.2	⊢ 𝑅 Se 𝐴
3		frins.3	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) [ 𝑧 / 𝑦 ] 𝜑 → 𝜑 ) )
4	3	frinsg	⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ 𝐴 𝜑 )
5	1 2 4	mp2an	⊢ ∀ 𝑦 ∈ 𝐴 𝜑
6	5	rspec	⊢ ( 𝑦 ∈ 𝐴 → 𝜑 )