Theorem deceq12i 9409

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 9407	. 2 ⊢ ;𝐴𝐶 = ;𝐵𝐶
3		deceq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	deceq2i 9408	. 2 ⊢ ;𝐵𝐶 = ;𝐵𝐷
5	2, 4	eqtri 2209	1 ⊢ ;𝐴𝐶 = ;𝐵𝐷

Syntax hints:   = wceq 1363  ;cdc 9401

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-rex 2473  df-v 2753  df-un 3147  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-iota 5192  df-fv 5238  df-ov 5893  df-dec 9402

This theorem is referenced by:  11multnc  9468