Theorem sbeqalb

Description: Theorem *14.121 in WhiteheadRussell p. 185. (Contributed by Andrew Salmon, 28-Jun-2011) (Proof shortened by Wolf Lammen, 9-May-2013)

	Ref	Expression
Assertion	sbeqalb	$$⊢A\in V\to \left(\left(\forall x\left(\phi \leftrightarrow x=A\right)\wedge \forall x\left(\phi \leftrightarrow x=B\right)\right)\to A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		bibi1	$$⊢\left(\phi \leftrightarrow x=A\right)\to \left(\left(\phi \leftrightarrow x=B\right)\leftrightarrow \left(x=A\leftrightarrow x=B\right)\right)$$
2	1	biimpa	$$⊢\left(\left(\phi \leftrightarrow x=A\right)\wedge \left(\phi \leftrightarrow x=B\right)\right)\to \left(x=A\leftrightarrow x=B\right)$$
3	2	biimpd	$$⊢\left(\left(\phi \leftrightarrow x=A\right)\wedge \left(\phi \leftrightarrow x=B\right)\right)\to \left(x=A\to x=B\right)$$
4	3	alanimi	$$⊢\left(\forall x\left(\phi \leftrightarrow x=A\right)\wedge \forall x\left(\phi \leftrightarrow x=B\right)\right)\to \forall x\left(x=A\to x=B\right)$$
5		sbceqal	$$⊢A\in V\to \left(\forall x\left(x=A\to x=B\right)\to A=B\right)$$
6	4 5	syl5	$$⊢A\in V\to \left(\left(\forall x\left(\phi \leftrightarrow x=A\right)\wedge \forall x\left(\phi \leftrightarrow x=B\right)\right)\to A=B\right)$$