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Metamath Proof Explorer

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