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 )