Theorem wfis2g

Description: Well-Ordered Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011)

	Ref	Expression
Hypotheses	wfis2g.1	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
	wfis2g.2	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
Assertion	wfis2g	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		wfis2g.1	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
2		wfis2g.2	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
3		nfv	⊢ Ⅎ 𝑦 𝜓
4	3 1 2	wfis2fg	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ 𝐴 𝜑 )