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Metamath Proof Explorer

Theorem wfis2fg

Description: Well-Ordered Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011) (Proof shortened by Scott Fenton, 17-Nov-2024)

		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		wefr	⊢ ( 𝑅 We 𝐴 → 𝑅 Fr 𝐴 )
5	4	adantr	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Fr 𝐴 )
6		weso	⊢ ( 𝑅 We 𝐴 → 𝑅 Or 𝐴 )
7		sopo	⊢ ( 𝑅 Or 𝐴 → 𝑅 Po 𝐴 )
8	6 7	syl	⊢ ( 𝑅 We 𝐴 → 𝑅 Po 𝐴 )
9	8	adantr	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Po 𝐴 )
10		simpr	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Se 𝐴 )
11	3 1 2	frpoins2fg	⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ 𝐴 𝜑 )
12	5 9 10 11	syl3anc	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ 𝐴 𝜑 )