Theorem imaeq12d

Description: Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016)

	Ref	Expression
Hypotheses	imaeq1d.1	$$⊢\phi \to A=B$$
	imaeq12d.2	$$⊢\phi \to C=D$$
Assertion	imaeq12d	$$⊢\phi \to A\left[C\right]=B\left[D\right]$$

Proof

Step	Hyp	Ref	Expression
1		imaeq1d.1	$$⊢\phi \to A=B$$
2		imaeq12d.2	$$⊢\phi \to C=D$$
3	1	imaeq1d	$$⊢\phi \to A\left[C\right]=B\left[C\right]$$
4	2	imaeq2d	$$⊢\phi \to B\left[C\right]=B\left[D\right]$$
5	3 4	eqtrd	$$⊢\phi \to A\left[C\right]=B\left[D\right]$$