Theorem s4eqd 14874

Description: Equality theorem for a length 4 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Hypotheses

Ref	Expression
s2eqd.1	⊢ (𝜑 → 𝐴 = 𝑁)
s2eqd.2	⊢ (𝜑 → 𝐵 = 𝑂)
s3eqd.3	⊢ (𝜑 → 𝐶 = 𝑃)
s4eqd.4	⊢ (𝜑 → 𝐷 = 𝑄)

Assertion

Ref	Expression
s4eqd	⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ = ⟨“𝑁𝑂𝑃𝑄”⟩)

Proof of Theorem s4eqd

Step	Hyp	Ref	Expression
1		s2eqd.1	. . . 4 ⊢ (𝜑 → 𝐴 = 𝑁)
2		s2eqd.2	. . . 4 ⊢ (𝜑 → 𝐵 = 𝑂)
3		s3eqd.3	. . . 4 ⊢ (𝜑 → 𝐶 = 𝑃)
4	1, 2, 3	s3eqd 14873	. . 3 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝑁𝑂𝑃”⟩)
5		s4eqd.4	. . . 4 ⊢ (𝜑 → 𝐷 = 𝑄)
6	5	s1eqd 14609	. . 3 ⊢ (𝜑 → ⟨“𝐷”⟩ = ⟨“𝑄”⟩)
7	4, 6	oveq12d 7442	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩) = (⟨“𝑁𝑂𝑃”⟩ ++ ⟨“𝑄”⟩))
8		df-s4 14859	. 2 ⊢ ⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
9		df-s4 14859	. 2 ⊢ ⟨“𝑁𝑂𝑃𝑄”⟩ = (⟨“𝑁𝑂𝑃”⟩ ++ ⟨“𝑄”⟩)
10	7, 8, 9	3eqtr4g 2791	1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ = ⟨“𝑁𝑂𝑃𝑄”⟩)

Syntax hints:   → wi 4   = wceq 1534  (class class class)co 7424   ++ cconcat 14578  ⟨“cs1 14603  ⟨“cs3 14851  ⟨“cs4 14852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-iota 6506  df-fv 6562  df-ov 7427  df-s1 14604  df-s2 14857  df-s3 14858  df-s4 14859

This theorem is referenced by:  s5eqd  14875