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Metamath Proof Explorer

Theorem speivw

Description: Version of spei with a disjoint variable condition, which does not require ax-13 (neither ax-7 nor ax-12 ). (Contributed by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	speivw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
	speivw.2	⊢ 𝜓
Assertion	speivw	⊢ ∃ 𝑥 𝜑

Proof

Step	Hyp	Ref	Expression
1		speivw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		speivw.2	⊢ 𝜓
3	1	biimprd	⊢ ( 𝑥 = 𝑦 → ( 𝜓 → 𝜑 ) )
4	3 2	speiv	⊢ ∃ 𝑥 𝜑