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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 )