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Metamath Proof Explorer

Theorem spcedv

Description: Existential specialization, using implicit substitution, deduction version. (Contributed by RP, 12-Aug-2020) (Revised by AV, 16-Aug-2024)

	Ref	Expression
Hypotheses	spcedv.1	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	spcedv.2	⊢ ( 𝜑 → 𝜒 )
	spcedv.3	⊢ ( 𝑥 = 𝑋 → ( 𝜓 ↔ 𝜒 ) )
Assertion	spcedv	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		spcedv.1	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
2		spcedv.2	⊢ ( 𝜑 → 𝜒 )
3		spcedv.3	⊢ ( 𝑥 = 𝑋 → ( 𝜓 ↔ 𝜒 ) )
4	3	spcegv	⊢ ( 𝑋 ∈ 𝑉 → ( 𝜒 → ∃ 𝑥 𝜓 ) )
5	1 2 4	sylc	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )