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Metamath Proof Explorer

Theorem setinds2f

Description: _E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011) (Revised by Mario Carneiro, 11-Dec-2016)

		Ref	Expression
	Hypotheses	setinds2f.1	$⊢ Ⅎ x ψ$
		setinds2f.2	$⊢ x = y \to (φ \leftrightarrow ψ)$
		setinds2f.3	$⊢ \forall y \in x ψ \to φ$
	Assertion	setinds2f	$⊢ φ$

Proof

Step	Hyp	Ref	Expression
1		setinds2f.1	$⊢ Ⅎ x ψ$
2		setinds2f.2	$⊢ x = y \to (φ \leftrightarrow ψ)$
3		setinds2f.3	$⊢ \forall y \in x ψ \to φ$
4		sbsbc	$⊢ [y/ x] φ \leftrightarrow \underset{˙}{[} y / x \underset{˙}{]} φ$
5	1 2	sbiev	$⊢ [y/ x] φ \leftrightarrow ψ$
6	4 5	bitr3i	$⊢ \underset{˙}{[} y / x \underset{˙}{]} φ \leftrightarrow ψ$
7	6	ralbii	$⊢ \forall y \in x \underset{˙}{[} y / x \underset{˙}{]} φ \leftrightarrow \forall y \in x ψ$
8	7 3	sylbi	$⊢ \forall y \in x \underset{˙}{[} y / x \underset{˙}{]} φ \to φ$
9	8	setinds	$⊢ φ$