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Theorem s2eqd 14428
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 14158 . . 3 (𝜑 → ⟨“𝐴”⟩ = ⟨“𝑁”⟩)
3 s2eqd.2 . . . 4 (𝜑𝐵 = 𝑂)
43s1eqd 14158 . . 3 (𝜑 → ⟨“𝐵”⟩ = ⟨“𝑂”⟩)
52, 4oveq12d 7231 . 2 (𝜑 → (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩) = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩))
6 df-s2 14413 . 2 ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
7 df-s2 14413 . 2 ⟨“𝑁𝑂”⟩ = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩)
85, 6, 73eqtr4g 2803 1 (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  (class class class)co 7213   ++ cconcat 14125  ⟨“cs1 14152  ⟨“cs2 14406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-s1 14153  df-s2 14413
This theorem is referenced by:  s3eqd  14429  swrds2m  14506  wrdl2exs2  14511  swrd2lsw  14517  efgi  19109  efgi0  19110  efgi1  19111  efgtf  19112  efgtval  19113  efgval2  19114  frgpuplem  19162  2clwwlk2clwwlklem  28429  wrdt2ind  30945
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