Theorem s2eqd 14816

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 14555	. . 3 ⊢ (𝜑 → ⟨“𝐴”⟩ = ⟨“𝑁”⟩)
3		s2eqd.2	. . . 4 ⊢ (𝜑 → 𝐵 = 𝑂)
4	3	s1eqd 14555	. . 3 ⊢ (𝜑 → ⟨“𝐵”⟩ = ⟨“𝑂”⟩)
5	2, 4	oveq12d 7374	. 2 ⊢ (𝜑 → (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩) = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩))
6		df-s2 14801	. 2 ⊢ ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
7		df-s2 14801	. 2 ⊢ ⟨“𝑁𝑂”⟩ = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩)
8	5, 6, 7	3eqtr4g 2799	1 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)

Syntax hints:   → wi 4   = wceq 1547  (class class class)co 7356   ++ cconcat 14523  ⟨“cs1 14549  ⟨“cs2 14794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359  df-s1 14550  df-s2 14801

This theorem is referenced by:  s3eqd  14817  swrds2m  14894  wrdl2exs2  14899  swrd2lsw  14905  efgi  19685  efgi0  19686  efgi1  19687  efgtf  19688  efgtval  19689  efgval2  19690  frgpuplem  19738  2clwwlk2clwwlklem  30434  wrdt2ind  33032  elrgspnsubrunlem1  33328  elrgspnsubrun  33330