Theorem s4eqd 13610

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 13609	. . 3 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝑁𝑂𝑃”⟩)
5		s4eqd.4	. . . 4 ⊢ (𝜑 → 𝐷 = 𝑄)
6	5	s1eqd 13381	. . 3 ⊢ (𝜑 → ⟨“𝐷”⟩ = ⟨“𝑄”⟩)
7	4, 6	oveq12d 6668	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩) = (⟨“𝑁𝑂𝑃”⟩ ++ ⟨“𝑄”⟩))
8		df-s4 13595	. 2 ⊢ ⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
9		df-s4 13595	. 2 ⊢ ⟨“𝑁𝑂𝑃𝑄”⟩ = (⟨“𝑁𝑂𝑃”⟩ ++ ⟨“𝑄”⟩)
10	7, 8, 9	3eqtr4g 2681	1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ = ⟨“𝑁𝑂𝑃𝑄”⟩)

Syntax hints:   → wi 4   = wceq 1483  (class class class)co 6650   ++ cconcat 13293  ⟨“cs1 13294  ⟨“cs3 13587  ⟨“cs4 13588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-s1 13302  df-s2 13593  df-s3 13594  df-s4 13595

This theorem is referenced by:  s5eqd  13611