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Metamath Proof Explorer

Theorem spc2ev

Description: Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995)

Ref Expression
Hypotheses spc2ev.1 ⊢ A ∈ V
spc2ev.2 ⊢ B ∈ V
spc2ev.3 ⊢ x = A ∧ y = B → φ ↔ ψ
Assertion spc2ev ⊢ ψ → ∃ x ∃ y φ

Proof

Step Hyp Ref Expression
1 spc2ev.1 ⊢ A ∈ V
2 spc2ev.2 ⊢ B ∈ V
3 spc2ev.3 ⊢ x = A ∧ y = B → φ ↔ ψ
4 3 spc2egv ⊢ A ∈ V ∧ B ∈ V → ψ → ∃ x ∃ y φ
5 1 2 4 mp2an ⊢ ψ → ∃ x ∃ y φ