Metamath Proof Explorer

Theorem vtocl3g

Description: Implicit substitution of a class for a setvar variable. Version of vtocl3gf with disjoint variable conditions instead of nonfreeness hypotheses, requiring fewer axioms. (Contributed by Gino Giotto, 3-Oct-2024)

		Ref	Expression
	Hypotheses	vtocl3g.1	$⊢ x = A \to (φ \leftrightarrow ψ)$
		vtocl3g.2	$⊢ y = B \to (ψ \leftrightarrow χ)$
		vtocl3g.3	$⊢ z = C \to (χ \leftrightarrow θ)$
		vtocl3g.4	$⊢ φ$
	Assertion	vtocl3g	$⊢ (A \in V \land B \in W \land C \in X) \to θ$

Proof

Step	Hyp	Ref	Expression
1		vtocl3g.1	$⊢ x = A \to (φ \leftrightarrow ψ)$
2		vtocl3g.2	$⊢ y = B \to (ψ \leftrightarrow χ)$
3		vtocl3g.3	$⊢ z = C \to (χ \leftrightarrow θ)$
4		vtocl3g.4	$⊢ φ$
5		elex	$⊢ A \in V \to A \in V$
6	2	imbi2d	$⊢ y = B \to ((A \in V \to ψ) \leftrightarrow (A \in V \to χ))$
7	3	imbi2d	$⊢ z = C \to ((A \in V \to χ) \leftrightarrow (A \in V \to θ))$
8	1 4	vtoclg	$⊢ A \in V \to ψ$
9	6 7 8	vtocl2g	$⊢ (B \in W \land C \in X) \to (A \in V \to θ)$
10	5 9	mpan9	$⊢ (A \in V \land (B \in W \land C \in X)) \to θ$
11	10	3impb	$⊢ (A \in V \land B \in W \land C \in X) \to θ$