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Metamath Proof Explorer

Theorem vtoclef

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993)

Ref Expression
Hypotheses vtoclef.1 ⊢ Ⅎ x φ
vtoclef.2 ⊢ A ∈ V
vtoclef.3 ⊢ x = A → φ
Assertion vtoclef ⊢ φ

Proof

Step Hyp Ref Expression
1 vtoclef.1 ⊢ Ⅎ x φ
2 vtoclef.2 ⊢ A ∈ V
3 vtoclef.3 ⊢ x = A → φ
4 2 isseti ⊢ ∃ x x = A
5 1 3 exlimi ⊢ ∃ x x = A → φ
6 4 5 ax-mp ⊢ φ