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Metamath Proof Explorer

Theorem rspcedv

Description: Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007) (Revised by Mario Carneiro, 4-Jan-2017)

	Ref	Expression
Hypotheses	rspcdv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
	rspcdv.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	rspcedv	⊢ ( 𝜑 → ( 𝜒 → ∃ 𝑥 ∈ 𝐵 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		rspcdv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2		rspcdv.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3	2	biimprd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜒 → 𝜓 ) )
4	1 3	rspcimedv	⊢ ( 𝜑 → ( 𝜒 → ∃ 𝑥 ∈ 𝐵 𝜓 ) )