Theorem frins2f

Description: Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 7-Feb-2011) (Revised by Mario Carneiro, 11-Dec-2016)

	Ref	Expression
Hypotheses	frins2f.1	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
	frins2f.2	⊢ Ⅎ 𝑦 𝜓
	frins2f.3	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
Assertion	frins2f	⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		frins2f.1	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
2		frins2f.2	⊢ Ⅎ 𝑦 𝜓
3		frins2f.3	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
4		sbsbc	⊢ ( [ 𝑧 / 𝑦 ] 𝜑 ↔ [ 𝑧 / 𝑦 ] 𝜑 )
5	2 3	sbiev	⊢ ( [ 𝑧 / 𝑦 ] 𝜑 ↔ 𝜓 )
6	4 5	bitr3i	⊢ ( [ 𝑧 / 𝑦 ] 𝜑 ↔ 𝜓 )
7	6	ralbii	⊢ ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) [ 𝑧 / 𝑦 ] 𝜑 ↔ ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 )
8	7 1	syl5bi	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) [ 𝑧 / 𝑦 ] 𝜑 → 𝜑 ) )
9	8	frinsg	⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ 𝐴 𝜑 )