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Theorem s2eqd 14218
 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 13948 . . 3 (𝜑 → ⟨“𝐴”⟩ = ⟨“𝑁”⟩)
3 s2eqd.2 . . . 4 (𝜑𝐵 = 𝑂)
43s1eqd 13948 . . 3 (𝜑 → ⟨“𝐵”⟩ = ⟨“𝑂”⟩)
52, 4oveq12d 7153 . 2 (𝜑 → (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩) = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩))
6 df-s2 14203 . 2 ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
7 df-s2 14203 . 2 ⟨“𝑁𝑂”⟩ = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩)
85, 6, 73eqtr4g 2858 1 (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  (class class class)co 7135   ++ cconcat 13915  ⟨“cs1 13942  ⟨“cs2 14196 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-s1 13943  df-s2 14203 This theorem is referenced by:  s3eqd  14219  swrds2m  14296  wrdl2exs2  14301  swrd2lsw  14307  efgi  18840  efgi0  18841  efgi1  18842  efgtf  18843  efgtval  18844  efgval2  18845  frgpuplem  18893  2clwwlk2clwwlklem  28138  wrdt2ind  30660
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