Theorem oteq3d 4818

Description: Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypothesis

Ref	Expression
oteq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
oteq3d	⊢ (𝜑 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)

Proof of Theorem oteq3d

Step	Hyp	Ref	Expression
1		oteq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		oteq3 4815	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
3	1, 2	syl 17	1 ⊢ (𝜑 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)

Syntax hints:   → wi 4   = wceq 1547  ⟨cotp 4563

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-ot 4564

This theorem is referenced by:  oteq123d  4819  idafval  18015  coafval  18022  arwlid  18030  arwrid  18031  arwass  18032  efgi  19685  efgtf  19688  efgtval  19689  efgval2  19690  mapdh6bN  42229  mapdh6cN  42230  mapdh6dN  42231  mapdh6gN  42234  hdmap1l6b  42303  hdmap1l6c  42304  hdmap1l6d  42305  hdmap1l6g  42308