Theorem eqeq1dALT

Description: Alternate proof of eqeq1d , shorter but requiring ax-12 . (Contributed by NM, 27-Dec-1993) (Revised by Wolf Lammen, 19-Nov-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	eqeq1d.1	$$⊢\phi \to A=B$$
Assertion	eqeq1dALT	$$⊢\phi \to \left(A=C\leftrightarrow B=C\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqeq1d.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	3	19.21bi	$$⊢\phi \to \left(x\in A\leftrightarrow x\in B\right)$$
5	4	bibi1d	$$⊢\phi \to \left(\left(x\in A\leftrightarrow x\in C\right)\leftrightarrow \left(x\in B\leftrightarrow x\in C\right)\right)$$
6	5	albidv	$$⊢\phi \to \left(\forall x\left(x\in A\leftrightarrow x\in C\right)\leftrightarrow \forall x\left(x\in B\leftrightarrow x\in C\right)\right)$$
7		dfcleq	$$⊢A=C\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in C\right)$$
8		dfcleq	$$⊢B=C\leftrightarrow \forall x\left(x\in B\leftrightarrow x\in C\right)$$
9	6 7 8	3bitr4g	$$⊢\phi \to \left(A=C\leftrightarrow B=C\right)$$