Theorem preq12d 3754

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

Hypotheses

Ref	Expression
preq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
preq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
preq12d	⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷})

Proof of Theorem preq12d

Step	Hyp	Ref	Expression
1		preq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		preq12d.2	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
3		preq12 3748	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷})
4	1, 2, 3	syl2anc 411	1 ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷})

Syntax hints:   → wi 4   = wceq 1395  {cpr 3668

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674

This theorem is referenced by:  opeq1  3860  opeq2  3861  xrminrecl  11827  xrminadd  11829  prdsval  13349  xpsfval  13424  xpsval  13428  ring1  14065  xmetxp  15224  xmetxpbl  15225  txmetcnp  15235  hovera  15364  hoverb  15365  hoverlt1  15366  hovergt0  15367  ivthdich  15370  wkslem1  16131  wkslem2  16132  iswlk  16134  2wlklem  16185  isclwwlk  16203  clwwlkccatlem  16209  clwwlkccat  16210  clwwlkn2  16230  clwwlkext2edg  16231  umgr2cwwk2dif  16233  s2elclwwlknon2  16245  clwwlknonex2lem2  16247  clwwlknonex2  16248  eupthseg  16261