Theorem preq1d 3653

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

Hypothesis

Ref	Expression
preq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
preq1d	|- ( ph -> { A , C } = { B , C } )

Proof of Theorem preq1d

Step	Hyp	Ref	Expression
1		preq1d.1	. 2 |- ( ph -> A = B )
2		preq1 3647	. 2 |- ( A = B -> { A , C } = { B , C } )
3	1, 2	syl 14	1 |- ( ph -> { A , C } = { B , C } )

Syntax hints:   -> wi 4   = wceq 1342   {cpr 3571

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115  df-sn 3576  df-pr 3577

This theorem is referenced by:  xrbdtri  11203  bdmetval  13041