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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	$⊢ y = z \to (φ \leftrightarrow ψ)$
		wfis2fg.3	$⊢ y \in A \to (\forall z \in Pred (R, A, y) ψ \to φ)$
	Assertion	wfis2fg	$⊢ (R We A \land R Se A) \to \forall y \in A φ$

Proof

Step	Hyp	Ref	Expression
1		wfis2fg.1	$⊢ Ⅎ y ψ$
2		wfis2fg.2	$⊢ y = z \to (φ \leftrightarrow ψ)$
3		wfis2fg.3	$⊢ y \in A \to (\forall z \in Pred (R, A, y) ψ \to φ)$
4		wefr	$⊢ R We A \to R Fr A$
5	4	adantr	$⊢ (R We A \land R Se A) \to R Fr A$
6		weso	$⊢ R We A \to R Or A$
7		sopo	$⊢ R Or A \to R Po A$
8	6 7	syl	$⊢ R We A \to R Po A$
9	8	adantr	$⊢ (R We A \land R Se A) \to R Po A$
10		simpr	$⊢ (R We A \land R Se A) \to R Se A$
11	3 1 2	frpoins2fg	$⊢ (R Fr A \land R Po A \land R Se A) \to \forall y \in A φ$
12	5 9 10 11	syl3anc	$⊢ (R We A \land R Se A) \to \forall y \in A φ$