Theorem rexraleqim

Description: Statement following from existence and generalization with equality. (Contributed by AV, 9-Feb-2019)

	Ref	Expression
Hypotheses	rexraleqim.1	$$⊢x=z\to \left(\psi \leftrightarrow \phi \right)$$
	rexraleqim.2	$$⊢z=Y\to \left(\phi \leftrightarrow \theta \right)$$
Assertion	rexraleqim	$$⊢\left(\exists z\in A\phi \wedge \forall x\in A\left(\psi \to x=Y\right)\right)\to \theta$$

Proof

Step	Hyp	Ref	Expression
1		rexraleqim.1	$$⊢x=z\to \left(\psi \leftrightarrow \phi \right)$$
2		rexraleqim.2	$$⊢z=Y\to \left(\phi \leftrightarrow \theta \right)$$
3		eqeq1	$$⊢x=z\to \left(x=Y\leftrightarrow z=Y\right)$$
4	1 3	imbi12d	$$⊢x=z\to \left(\left(\psi \to x=Y\right)\leftrightarrow \left(\phi \to z=Y\right)\right)$$
5	4	rspcva	$$⊢\left(z\in A\wedge \forall x\in A\left(\psi \to x=Y\right)\right)\to \left(\phi \to z=Y\right)$$
6	2	biimpd	$$⊢z=Y\to \left(\phi \to \theta \right)$$
7	5 6	syli	$$⊢\left(z\in A\wedge \forall x\in A\left(\psi \to x=Y\right)\right)\to \left(\phi \to \theta \right)$$
8	7	impancom	$$⊢\left(z\in A\wedge \phi \right)\to \left(\forall x\in A\left(\psi \to x=Y\right)\to \theta \right)$$
9	8	rexlimiva	$$⊢\exists z\in A\phi \to \left(\forall x\in A\left(\psi \to x=Y\right)\to \theta \right)$$
10	9	imp	$$⊢\left(\exists z\in A\phi \wedge \forall x\in A\left(\psi \to x=Y\right)\right)\to \theta$$