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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