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	$$⊢\left(A\in B\wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\to \exists x\in B\phi$$

Proof

Step	Hyp	Ref	Expression
1		dfsbcq2	$$⊢y=A\to \left(\left[y/x\right]\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
2	1	rspcev	$$⊢\left(A\in B\wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\to \exists y\in B\left[y/x\right]\phi$$
3		cbvrexsvw	$$⊢\exists x\in B\phi \leftrightarrow \exists y\in B\left[y/x\right]\phi$$
4	2 3	sylibr	$$⊢\left(A\in B\wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\to \exists x\in B\phi$$