Theorem oteq123d 4780

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

Hypotheses

Ref	Expression
oteq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
oteq123d.2	⊢ (𝜑 → 𝐶 = 𝐷)
oteq123d.3	⊢ (𝜑 → 𝐸 = 𝐹)

Assertion

Ref	Expression
oteq123d	⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)

Proof of Theorem oteq123d

Step	Hyp	Ref	Expression
1		oteq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	oteq1d 4777	. 2 ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐶, 𝐸⟩)
3		oteq123d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	oteq2d 4778	. 2 ⊢ (𝜑 → ⟨𝐵, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐸⟩)
5		oteq123d.3	. . 3 ⊢ (𝜑 → 𝐸 = 𝐹)
6	5	oteq3d 4779	. 2 ⊢ (𝜑 → ⟨𝐵, 𝐷, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
7	2, 4, 6	3eqtrd 2837	1 ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)

Syntax hints:   → wi 4   = wceq 1538  ⟨cotp 4533

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-sn 4526  df-pr 4528  df-op 4532  df-ot 4534

This theorem is referenced by:  idaval  17310  coaval  17320  matval  21016  msrval  32898  mclsax  32929  elmpps  32933  mthmpps  32942  ackval0012  45103  ackval1012  45104  ackval2012  45105  ackval3012  45106