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Metamath Proof Explorer

Theorem spimevw

Description: Existential introduction, using implicit substitution. This is to spimew what spimvw is to spimw . Version of spimev and spimefv with an additional disjoint variable condition, using only Tarski's FOL axiom schemes. (Contributed by NM, 10-Jan-1993) (Revised by BJ, 17-Mar-2020)

		Ref	Expression
	Hypothesis	spimevw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
	Assertion	spimevw	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		spimevw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
2		ax-5	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
3	2 1	spimew	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )