Theorem preq12d 3677

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

Hypotheses

Ref	Expression
preq1d.1	|- ( ph -> A = B )
preq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
preq12d	|- ( ph -> { A , C } = { B , D } )

Proof of Theorem preq12d

Step	Hyp	Ref	Expression
1		preq1d.1	. 2 |- ( ph -> A = B )
2		preq12d.2	. 2 |- ( ph -> C = D )
3		preq12 3671	. 2 |- ( ( A = B /\ C = D ) -> { A , C } = { B , D } )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> { A , C } = { B , D } )

Syntax hints:   -> wi 4   = wceq 1353   {cpr 3593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599

This theorem is referenced by:  opeq1  3778  opeq2  3779  xrminrecl  11276  xrminadd  11278  ring1  13189  xmetxp  13900  xmetxpbl  13901  txmetcnp  13911