Theorem s2eqd 11263

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 11114	. . 3 ⊢ (𝜑 → ⟨“𝐴”⟩ = ⟨“𝑁”⟩)
3		s2eqd.2	. . . 4 ⊢ (𝜑 → 𝐵 = 𝑂)
4	3	s1eqd 11114	. . 3 ⊢ (𝜑 → ⟨“𝐵”⟩ = ⟨“𝑂”⟩)
5	2, 4	oveq12d 5987	. 2 ⊢ (𝜑 → (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩) = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩))
6		df-s2 11249	. 2 ⊢ ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
7		df-s2 11249	. 2 ⊢ ⟨“𝑁𝑂”⟩ = (⟨“𝑁”⟩ ++ ⟨“𝑂”⟩)
8	5, 6, 7	3eqtr4g 2265	1 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)

Syntax hints:   → wi 4   = wceq 1373  (class class class)co 5969   ++ cconcat 11086  ⟨“cs1 11109  ⟨“cs2 11242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2779  df-un 3179  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-br 4061  df-iota 5252  df-fv 5299  df-ov 5972  df-s1 11110  df-s2 11249

This theorem is referenced by:  s3eqd  11264