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Theorem s2eqd 13895
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 13575 . . 3 (𝜑 → ⟨“𝐴”⟩ = ⟨“𝑁”⟩)
3 s2eqd.2 . . . 4 (𝜑𝐵 = 𝑂)
43s1eqd 13575 . . 3 (𝜑 → ⟨“𝐵”⟩ = ⟨“𝑂”⟩)
52, 4oveq12d 6862 . 2 (𝜑 → (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩) = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩))
6 df-s2 13880 . 2 ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
7 df-s2 13880 . 2 ⟨“𝑁𝑂”⟩ = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩)
85, 6, 73eqtr4g 2824 1 (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  (class class class)co 6844   ++ cconcat 13544  ⟨“cs1 13569  ⟨“cs2 13873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-iota 6033  df-fv 6078  df-ov 6847  df-s1 13570  df-s2 13880
This theorem is referenced by:  s3eqd  13896  swrds2m  13973  wrdl2exs2  13978  swrd2lsw  13984  efgi  18399  efgi0  18400  efgi1  18401  efgtf  18402  efgtval  18403  efgval2  18404  frgpuplem  18454  2clwwlk2clwwlklem  27634
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