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Theorem s2eqd 14902
Description: Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
s2eqd.1 (𝜑𝐴 = 𝑁)
s2eqd.2 (𝜑𝐵 = 𝑂)
Assertion
Ref Expression
s2eqd (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)

Proof of Theorem s2eqd
StepHypRef Expression
1 s2eqd.1 . . . 4 (𝜑𝐴 = 𝑁)
21s1eqd 14639 . . 3 (𝜑 → ⟨“𝐴”⟩ = ⟨“𝑁”⟩)
3 s2eqd.2 . . . 4 (𝜑𝐵 = 𝑂)
43s1eqd 14639 . . 3 (𝜑 → ⟨“𝐵”⟩ = ⟨“𝑂”⟩)
52, 4oveq12d 7449 . 2 (𝜑 → (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩) = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩))
6 df-s2 14887 . 2 ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
7 df-s2 14887 . 2 ⟨“𝑁𝑂”⟩ = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩)
85, 6, 73eqtr4g 2802 1 (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  (class class class)co 7431   ++ cconcat 14608  ⟨“cs1 14633  ⟨“cs2 14880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-s1 14634  df-s2 14887
This theorem is referenced by:  s3eqd  14903  swrds2m  14980  wrdl2exs2  14985  swrd2lsw  14991  efgi  19737  efgi0  19738  efgi1  19739  efgtf  19740  efgtval  19741  efgval2  19742  frgpuplem  19790  2clwwlk2clwwlklem  30365  wrdt2ind  32938  elrgspnsubrunlem1  33251  elrgspnsubrun  33253
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