Theorem oteq2d 4778

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 4775	. 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 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:  oteq123d  4780  mapdh9a  39085  mapdh9aOLDN  39086  hdmap1eulem  39118  hdmap1eulemOLDN  39119  hdmapffval  39122  hdmapfval  39123  hdmapval2  39128