Theorem eleq2d

Description: Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019)

	Ref	Expression
Hypothesis	eleq1d.1	$$⊢\phi \to A=B$$
Assertion	eleq2d	$$⊢\phi \to \left(C\in A\leftrightarrow C\in B\right)$$

Proof

Step	Hyp	Ref	Expression
1		eleq1d.1	$$⊢\phi \to A=B$$
2		dfcleq	$$⊢A=B\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in B\right)$$
3	1 2	sylib	$$⊢\phi \to \forall x\left(x\in A\leftrightarrow x\in B\right)$$
4		anbi2	$$⊢\left(x\in A\leftrightarrow x\in B\right)\to \left(\left(x=C\wedge x\in A\right)\leftrightarrow \left(x=C\wedge x\in B\right)\right)$$
5	4	alexbii	$$⊢\forall x\left(x\in A\leftrightarrow x\in B\right)\to \left(\exists x\left(x=C\wedge x\in A\right)\leftrightarrow \exists x\left(x=C\wedge x\in B\right)\right)$$
6	3 5	syl	$$⊢\phi \to \left(\exists x\left(x=C\wedge x\in A\right)\leftrightarrow \exists x\left(x=C\wedge x\in B\right)\right)$$
7		dfclel	$$⊢C\in A\leftrightarrow \exists x\left(x=C\wedge x\in A\right)$$
8		dfclel	$$⊢C\in B\leftrightarrow \exists x\left(x=C\wedge x\in B\right)$$
9	6 7 8	3bitr4g	$$⊢\phi \to \left(C\in A\leftrightarrow C\in B\right)$$