Theorem s2eqd 14818

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 7429	. 2 ⊢ (𝜑 → (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩) = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩))
6		df-s2 14803	. 2 ⊢ ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
7		df-s2 14803	. 2 ⊢ ⟨“𝑁𝑂”⟩ = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩)
8	5, 6, 7	3eqtr4g 2795	1 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)

Syntax hints:   → wi 4   = wceq 1539  (class class class)co 7411   ++ cconcat 14524  ⟨“cs1 14549  ⟨“cs2 14796

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-s1 14550  df-s2 14803

This theorem is referenced by:  s3eqd  14819  swrds2m  14896  wrdl2exs2  14901  swrd2lsw  14907  efgi  19628  efgi0  19629  efgi1  19630  efgtf  19631  efgtval  19632  efgval2  19633  frgpuplem  19681  2clwwlk2clwwlklem  29866  wrdt2ind  32384