Theorem oteq123d 4417

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 4414	. 2 ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐶, 𝐸⟩)
3		oteq123d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	oteq2d 4415	. 2 ⊢ (𝜑 → ⟨𝐵, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐸⟩)
5		oteq123d.3	. . 3 ⊢ (𝜑 → 𝐸 = 𝐹)
6	5	oteq3d 4416	. 2 ⊢ (𝜑 → ⟨𝐵, 𝐷, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
7	2, 4, 6	3eqtrd 2660	1 ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)

Syntax hints:   → wi 4   = wceq 1483  ⟨cotp 4185

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-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-ot 4186

This theorem is referenced by:  idaval  16708  coaval  16718  matval  20217  msrval  31435  mclsax  31466  elmpps  31470  mthmpps  31479