Theorem seqeq123d

Description: Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013)

	Ref	Expression
Hypotheses	seqeq123d.1	$$⊢\phi \to M=N$$
	seqeq123d.2	$$⊢\phi \to \underset{\dot{}}{+}=Q$$
	seqeq123d.3	$$⊢\phi \to F=G$$
Assertion	seqeq123d	$$⊢\phi \to seqM\left(\underset{\dot{}}{+},F\right)=seqN\left(Q,G\right)$$

Proof

Step	Hyp	Ref	Expression
1		seqeq123d.1	$$⊢\phi \to M=N$$
2		seqeq123d.2	$$⊢\phi \to \underset{\dot{}}{+}=Q$$
3		seqeq123d.3	$$⊢\phi \to F=G$$
4	1	seqeq1d	$$⊢\phi \to seqM\left(\underset{\dot{}}{+},F\right)=seqN\left(\underset{\dot{}}{+},F\right)$$
5	2	seqeq2d	$$⊢\phi \to seqN\left(\underset{\dot{}}{+},F\right)=seqN\left(Q,F\right)$$
6	3	seqeq3d	$$⊢\phi \to seqN\left(Q,F\right)=seqN\left(Q,G\right)$$
7	4 5 6	3eqtrd	$$⊢\phi \to seqM\left(\underset{\dot{}}{+},F\right)=seqN\left(Q,G\right)$$