Theorem moi

Description: Equality implied by "at most one". (Contributed by NM, 18-Feb-2006)

	Ref	Expression
Hypotheses	moi.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	moi.2	$$⊢x=B\to \left(\phi \leftrightarrow \chi \right)$$
Assertion	moi	$$⊢\left(\left(A\in C\wedge B\in D\right)\wedge {\exists }^{*}x\phi \wedge \left(\psi \wedge \chi \right)\right)\to A=B$$

Proof

Step	Hyp	Ref	Expression
1		moi.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2		moi.2	$$⊢x=B\to \left(\phi \leftrightarrow \chi \right)$$
3	1 2	mob	$$⊢\left(\left(A\in C\wedge B\in D\right)\wedge {\exists }^{*}x\phi \wedge \psi \right)\to \left(A=B\leftrightarrow \chi \right)$$
4	3	biimprd	$$⊢\left(\left(A\in C\wedge B\in D\right)\wedge {\exists }^{*}x\phi \wedge \psi \right)\to \left(\chi \to A=B\right)$$
5	4	3expia	$$⊢\left(\left(A\in C\wedge B\in D\right)\wedge {\exists }^{*}x\phi \right)\to \left(\psi \to \left(\chi \to A=B\right)\right)$$
6	5	impd	$$⊢\left(\left(A\in C\wedge B\in D\right)\wedge {\exists }^{*}x\phi \right)\to \left(\left(\psi \wedge \chi \right)\to A=B\right)$$
7	6	3impia	$$⊢\left(\left(A\in C\wedge B\in D\right)\wedge {\exists }^{*}x\phi \wedge \left(\psi \wedge \chi \right)\right)\to A=B$$