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Metamath Proof Explorer

Theorem vtoclg1f

Description: Version of vtoclgf with one nonfreeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-10 and ax-11 . (Contributed by BJ, 1-May-2019)

	Ref	Expression
Hypotheses	vtoclg1f.nf	$⊢ Ⅎ x ψ$
	vtoclg1f.maj	$⊢ x = A \to (φ \leftrightarrow ψ)$
	vtoclg1f.min	$⊢ φ$
Assertion	vtoclg1f	$⊢ A \in V \to ψ$

Proof

Step	Hyp	Ref	Expression
1		vtoclg1f.nf	$⊢ Ⅎ x ψ$
2		vtoclg1f.maj	$⊢ x = A \to (φ \leftrightarrow ψ)$
3		vtoclg1f.min	$⊢ φ$
4		elisset	$⊢ A \in V \to \exists x x = A$
5	3 2	mpbii	$⊢ x = A \to ψ$
6	1 5	exlimi	$⊢ \exists x x = A \to ψ$
7	4 6	syl	$⊢ A \in V \to ψ$