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Theorem s2eqd 13829
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 13592 . . 3 (𝜑 → ⟨“𝐴”⟩ = ⟨“𝑁”⟩)
3 s2eqd.2 . . . 4 (𝜑𝐵 = 𝑂)
43s1eqd 13592 . . 3 (𝜑 → ⟨“𝐵”⟩ = ⟨“𝑂”⟩)
52, 4oveq12d 6833 . 2 (𝜑 → (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩) = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩))
6 df-s2 13814 . 2 ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
7 df-s2 13814 . 2 ⟨“𝑁𝑂”⟩ = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩)
85, 6, 73eqtr4g 2820 1 (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  (class class class)co 6815   ++ cconcat 13500  ⟨“cs1 13501  ⟨“cs2 13807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-iota 6013  df-fv 6058  df-ov 6818  df-s1 13509  df-s2 13814
This theorem is referenced by:  s3eqd  13830  swrds2m  13907  wrdl2exs2  13912  swrd2lsw  13917  efgi  18353  efgi0  18354  efgi1  18355  efgtf  18356  efgtval  18357  efgval2  18358  frgpuplem  18406  2clwwlk2clwwlklem  27525
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