Theorem oteq2d 4910

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

Hypothesis

Ref	Expression
oteq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
oteq2d	⊢ (𝜑 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)

Proof of Theorem oteq2d

Step	Hyp	Ref	Expression
1		oteq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		oteq2 4907	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
3	1, 2	syl 17	1 ⊢ (𝜑 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)

Syntax hints:   → wi 4   = wceq 1537  ⟨cotp 4656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-ot 4657

This theorem is referenced by:  oteq123d  4912  mapdh9a  41746  mapdh9aOLDN  41747  hdmap1eulem  41779  hdmap1eulemOLDN  41780  hdmapffval  41783  hdmapfval  41784  hdmapval2  41789