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Metamath Proof Explorer

Theorem spimefv

Description: Version of spime with a disjoint variable condition, which does not require ax-13 . (Contributed by BJ, 31-May-2019)

Ref Expression
Hypotheses spimefv.1 ⊢ Ⅎ x φ
spimefv.2 ⊢ x = y → φ → ψ
Assertion spimefv ⊢ φ → ∃ x ψ

Proof

Step Hyp Ref Expression
1 spimefv.1 ⊢ Ⅎ x φ
2 spimefv.2 ⊢ x = y → φ → ψ
3 1 a1i ⊢ ⊤ → Ⅎ x φ
4 3 2 spimedv ⊢ ⊤ → φ → ∃ x ψ
5 4 mptru ⊢ φ → ∃ x ψ