Theorem oteq3d 4831

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 4828	. 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 1542  ⟨cotp 4576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-ot 4577

This theorem is referenced by:  oteq123d  4832  idafval  18018  coafval  18025  arwlid  18033  arwrid  18034  arwass  18035  efgi  19688  efgtf  19691  efgtval  19692  efgval2  19693  mapdh6bN  42200  mapdh6cN  42201  mapdh6dN  42202  mapdh6gN  42205  hdmap1l6b  42274  hdmap1l6c  42275  hdmap1l6d  42276  hdmap1l6g  42279