Theorem s2eqd 14788

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 14526	. . 3 ⊢ (𝜑 → ⟨“𝐴”⟩ = ⟨“𝑁”⟩)
3		s2eqd.2	. . . 4 ⊢ (𝜑 → 𝐵 = 𝑂)
4	3	s1eqd 14526	. . 3 ⊢ (𝜑 → ⟨“𝐵”⟩ = ⟨“𝑂”⟩)
5	2, 4	oveq12d 7371	. 2 ⊢ (𝜑 → (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩) = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩))
6		df-s2 14773	. 2 ⊢ ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
7		df-s2 14773	. 2 ⊢ ⟨“𝑁𝑂”⟩ = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩)
8	5, 6, 7	3eqtr4g 2789	1 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)

Syntax hints:   → wi 4   = wceq 1540  (class class class)co 7353   ++ cconcat 14495  ⟨“cs1 14520  ⟨“cs2 14766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-ov 7356  df-s1 14521  df-s2 14773

This theorem is referenced by:  s3eqd  14789  swrds2m  14866  wrdl2exs2  14871  swrd2lsw  14877  efgi  19616  efgi0  19617  efgi1  19618  efgtf  19619  efgtval  19620  efgval2  19621  frgpuplem  19669  2clwwlk2clwwlklem  30308  wrdt2ind  32908  elrgspnsubrunlem1  33200  elrgspnsubrun  33202