Theorem deceq12i 9768

Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)

Hypotheses

Ref	Expression
deceq1i.1	⊢ 𝐴 = 𝐵
deceq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
deceq12i	⊢ ;𝐴𝐶 = ;𝐵𝐷

Proof of Theorem deceq12i

Step	Hyp	Ref	Expression
1		deceq1i.1	. . 3 ⊢ 𝐴 = 𝐵
2	1	deceq1i 9766	. 2 ⊢ ;𝐴𝐶 = ;𝐵𝐶
3		deceq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	deceq2i 9767	. 2 ⊢ ;𝐵𝐶 = ;𝐵𝐷
5	2, 4	eqtri 2259	1 ⊢ ;𝐴𝐶 = ;𝐵𝐷

Syntax hints:   = wceq 1402  ;cdc 9760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-iota 5335  df-fv 5383  df-ov 6082  df-dec 9761

This theorem is referenced by:  11multnc  9827