Theorem preq12d 3795

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 3789	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷})
4	1, 2, 3	syl2anc 415	1 ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷})

Syntax hints:   → wi 4   = wceq 1402  {cpr 3709

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715

This theorem is referenced by:  opeq1  3902  opeq2  3903  xrminrecl  12022  xrminadd  12024  xpsfval  13652  prdsval  14156  xpsval  14184  ring1  14347  xmetxp  15591  xmetxpbl  15592  txmetcnp  15602  hovera  15731  hoverb  15732  hoverlt1  15733  hovergt0  15734  ivthdich  15737  wkslem1  16544  wkslem2  16545  iswlk  16547  2wlklem  16600  isclwwlk  16618  clwwlkccatlem  16624  clwwlkccat  16625  clwwlkn2  16645  clwwlkext2edg  16646  umgr2cwwk2dif  16648  s2elclwwlknon2  16660  clwwlknonex2lem2  16662  clwwlknonex2  16663  eupthseg  16676  eupth2lem3fi  16700