Theorem wfis2

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

	Ref	Expression
Hypotheses	wfis2.1	⊢ 𝑅 We 𝐴
	wfis2.2	⊢ 𝑅 Se 𝐴
	wfis2.3	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
	wfis2.4	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
Assertion	wfis2	⊢ ( 𝑦 ∈ 𝐴 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		wfis2.1	⊢ 𝑅 We 𝐴
2		wfis2.2	⊢ 𝑅 Se 𝐴
3		wfis2.3	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
4		wfis2.4	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
5	3 4	wfis2g	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ 𝐴 𝜑 )
6	1 2 5	mp2an	⊢ ∀ 𝑦 ∈ 𝐴 𝜑
7	6	rspec	⊢ ( 𝑦 ∈ 𝐴 → 𝜑 )