Theorem wfis2fg

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

	Ref	Expression
Hypotheses	wfis2fg.1	⊢ Ⅎ 𝑦 𝜓
	wfis2fg.2	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
	wfis2fg.3	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
Assertion	wfis2fg	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		wfis2fg.1	⊢ Ⅎ 𝑦 𝜓
2		wfis2fg.2	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
3		wfis2fg.3	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
4		sbsbc	⊢ ( [ 𝑧 / 𝑦 ] 𝜑 ↔ [ 𝑧 / 𝑦 ] 𝜑 )
5	1 2	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 3	syl5bi	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) [ 𝑧 / 𝑦 ] 𝜑 → 𝜑 ) )
9	8	wfisg	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ 𝐴 𝜑 )