Theorem rspesbca

Description: Existence form of rspsbca . (Contributed by NM, 29-Feb-2008) (Proof shortened by Mario Carneiro, 13-Oct-2016)

	Ref	Expression
Assertion	rspesbca	⊢ ( ( 𝐴 ∈ 𝐵 ∧ [ 𝐴 / 𝑥 ] 𝜑 ) → ∃ 𝑥 ∈ 𝐵 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		dfsbcq2	⊢ ( 𝑦 = 𝐴 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) )
2	1	rspcev	⊢ ( ( 𝐴 ∈ 𝐵 ∧ [ 𝐴 / 𝑥 ] 𝜑 ) → ∃ 𝑦 ∈ 𝐵 [ 𝑦 / 𝑥 ] 𝜑 )
3		cbvrexsvw	⊢ ( ∃ 𝑥 ∈ 𝐵 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 [ 𝑦 / 𝑥 ] 𝜑 )
4	2 3	sylibr	⊢ ( ( 𝐴 ∈ 𝐵 ∧ [ 𝐴 / 𝑥 ] 𝜑 ) → ∃ 𝑥 ∈ 𝐵 𝜑 )