Theorem eqeqan2d

Description: Implication of introducing a new equality. (Contributed by Peter Mazsa, 17-Apr-2019)

	Ref	Expression
Hypothesis	eqeqan2d.1	$$⊢\phi \to C=D$$
Assertion	eqeqan2d	$$⊢\left(A=B\wedge \phi \right)\to \left(A=C\leftrightarrow B=D\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqeqan2d.1	$$⊢\phi \to C=D$$
2		eqeq12	$$⊢\left(A=B\wedge C=D\right)\to \left(A=C\leftrightarrow B=D\right)$$
3	1 2	sylan2	$$⊢\left(A=B\wedge \phi \right)\to \left(A=C\leftrightarrow B=D\right)$$