Theorem preq1i 4665

Description: Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Hypothesis

Ref	Expression
preq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
preq1i	⊢ {𝐴, 𝐶} = {𝐵, 𝐶}

Proof of Theorem preq1i

Step	Hyp	Ref	Expression
1		preq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		preq1 4662	. 2 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
3	1, 2	ax-mp 5	1 ⊢ {𝐴, 𝐶} = {𝐵, 𝐶}

Syntax hints:   = wceq 1533  {cpr 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-un 3940  df-sn 4561  df-pr 4563

This theorem is referenced by:  funopg  6383  frcond1  28039  n4cyclfrgr  28064  disjdifprg2  30320