Theorem s2eqd 14869

Description: Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Hypotheses

Ref	Expression
s2eqd.1	⊢ (𝜑 → 𝐴 = 𝑁)
s2eqd.2	⊢ (𝜑 → 𝐵 = 𝑂)

Assertion

Ref	Expression
s2eqd	⊢ (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)

Proof of Theorem s2eqd

Step	Hyp	Ref	Expression
1		s2eqd.1	. . . 4 ⊢ (𝜑 → 𝐴 = 𝑁)
2	1	s1eqd 14608	. . 3 ⊢ (𝜑 → ⟨“𝐴”⟩ = ⟨“𝑁”⟩)
3		s2eqd.2	. . . 4 ⊢ (𝜑 → 𝐵 = 𝑂)
4	3	s1eqd 14608	. . 3 ⊢ (𝜑 → ⟨“𝐵”⟩ = ⟨“𝑂”⟩)
5	2, 4	oveq12d 7408	. 2 ⊢ (𝜑 → (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩) = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩))
6		df-s2 14854	. 2 ⊢ ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
7		df-s2 14854	. 2 ⊢ ⟨“𝑁𝑂”⟩ = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩)
8	5, 6, 7	3eqtr4g 2821	1 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)

Syntax hints:   → wi 4   = wceq 1559  (class class class)co 7390   ++ cconcat 14576  ⟨“cs1 14602  ⟨“cs2 14847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6471  df-fv 6523  df-ov 7393  df-s1 14603  df-s2 14854

This theorem is referenced by:  s3eqd  14870  swrds2m  14947  wrdl2exs2  14952  swrd2lsw  14958  efgi  19749  efgi0  19750  efgi1  19751  efgtf  19752  efgtval  19753  efgval2  19754  frgpuplem  19802  2clwwlk2clwwlklem  30504  wrdt2ind  33091  elrgspnsubrunlem1  33388  elrgspnsubrun  33390