Theorem oteq123d 3720

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 3717	. 2 ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐶, 𝐸⟩)
3		oteq123d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	oteq2d 3718	. 2 ⊢ (𝜑 → ⟨𝐵, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐸⟩)
5		oteq123d.3	. . 3 ⊢ (𝜑 → 𝐸 = 𝐹)
6	5	oteq3d 3719	. 2 ⊢ (𝜑 → ⟨𝐵, 𝐷, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
7	2, 4, 6	3eqtrd 2176	1 ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)

Syntax hints:   → wi 4   = wceq 1331  ⟨cotp 3531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-ot 3537

This theorem is referenced by: (None)