Theorem eqreu

Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypothesis	eqreu.1	$$⊢x=B\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	eqreu	$$⊢\left(B\in A\wedge \psi \wedge \forall x\in A\left(\phi \to x=B\right)\right)\to \exists !x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		eqreu.1	$$⊢x=B\to \left(\phi \leftrightarrow \psi \right)$$
2		ralbiim	$$⊢\forall x\in A\left(\phi \leftrightarrow x=B\right)\leftrightarrow \left(\forall x\in A\left(\phi \to x=B\right)\wedge \forall x\in A\left(x=B\to \phi \right)\right)$$
3	1	ceqsralv	$$⊢B\in A\to \left(\forall x\in A\left(x=B\to \phi \right)\leftrightarrow \psi \right)$$
4	3	anbi2d	$$⊢B\in A\to \left(\left(\forall x\in A\left(\phi \to x=B\right)\wedge \forall x\in A\left(x=B\to \phi \right)\right)\leftrightarrow \left(\forall x\in A\left(\phi \to x=B\right)\wedge \psi \right)\right)$$
5	2 4	bitrid	$$⊢B\in A\to \left(\forall x\in A\left(\phi \leftrightarrow x=B\right)\leftrightarrow \left(\forall x\in A\left(\phi \to x=B\right)\wedge \psi \right)\right)$$
6		reu6i	$$⊢\left(B\in A\wedge \forall x\in A\left(\phi \leftrightarrow x=B\right)\right)\to \exists !x\in A\phi$$
7	6	ex	$$⊢B\in A\to \left(\forall x\in A\left(\phi \leftrightarrow x=B\right)\to \exists !x\in A\phi \right)$$
8	5 7	sylbird	$$⊢B\in A\to \left(\left(\forall x\in A\left(\phi \to x=B\right)\wedge \psi \right)\to \exists !x\in A\phi \right)$$
9	8	3impib	$$⊢\left(B\in A\wedge \forall x\in A\left(\phi \to x=B\right)\wedge \psi \right)\to \exists !x\in A\phi$$
10	9	3com23	$$⊢\left(B\in A\wedge \psi \wedge \forall x\in A\left(\phi \to x=B\right)\right)\to \exists !x\in A\phi$$