Theorem reseq12d

Description: Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014)

	Ref	Expression
Hypotheses	reseqd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	reseqd.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	reseq12d	⊢ ( 𝜑 → ( 𝐴 ↾ 𝐶 ) = ( 𝐵 ↾ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		reseqd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		reseqd.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3	1	reseq1d	⊢ ( 𝜑 → ( 𝐴 ↾ 𝐶 ) = ( 𝐵 ↾ 𝐶 ) )
4	2	reseq2d	⊢ ( 𝜑 → ( 𝐵 ↾ 𝐶 ) = ( 𝐵 ↾ 𝐷 ) )
5	3 4	eqtrd	⊢ ( 𝜑 → ( 𝐴 ↾ 𝐶 ) = ( 𝐵 ↾ 𝐷 ) )