Theorem tpeq2d 4771

Description: Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Hypothesis

Ref	Expression
tpeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
tpeq2d	⊢ (𝜑 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})

Proof of Theorem tpeq2d

Step	Hyp	Ref	Expression
1		tpeq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		tpeq2 4768	. 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  {ctp 4652

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-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-sn 4649  df-pr 4651  df-tp 4653

This theorem is referenced by:  tpeq123d  4773  fntpb  7246  erngset  40757  erngset-rN  40765