Theorem wfis3

Description: Well-Ordered Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011)

	Ref	Expression
Hypotheses	wfis3.1	⊢ 𝑅 We 𝐴
	wfis3.2	⊢ 𝑅 Se 𝐴
	wfis3.3	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
	wfis3.4	⊢ ( 𝑦 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
	wfis3.5	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
Assertion	wfis3	⊢ ( 𝐵 ∈ 𝐴 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		wfis3.1	⊢ 𝑅 We 𝐴
2		wfis3.2	⊢ 𝑅 Se 𝐴
3		wfis3.3	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
4		wfis3.4	⊢ ( 𝑦 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
5		wfis3.5	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
6	1 2 3 5	wfis2	⊢ ( 𝑦 ∈ 𝐴 → 𝜑 )
7	4 6	vtoclga	⊢ ( 𝐵 ∈ 𝐴 → 𝜒 )