Theorem frins3

Description: Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	frins3.1	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
	frins3.2	⊢ ( 𝑦 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
	frins3.3	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
Assertion	frins3	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝐵 ∈ 𝐴 ) → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		frins3.1	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
2		frins3.2	⊢ ( 𝑦 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
3		frins3.3	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
4	3 1	frins2	⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ 𝐴 𝜑 )
5	2	rspcv	⊢ ( 𝐵 ∈ 𝐴 → ( ∀ 𝑦 ∈ 𝐴 𝜑 → 𝜒 ) )
6	4 5	mpan9	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝐵 ∈ 𝐴 ) → 𝜒 )