Theorem preq12 3802

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

Assertion

Ref	Expression
preq12	⊢ ((A = C ∧ B = D) → {A, B} = {C, D})

Proof of Theorem preq12

Step	Hyp	Ref	Expression
1		preq1 3800	. 2 ⊢ (A = C → {A, B} = {C, B})
2		preq2 3801	. 2 ⊢ (B = D → {C, B} = {C, D})
3	1, 2	sylan9eq 2405	1 ⊢ ((A = C ∧ B = D) → {A, B} = {C, D})

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  {cpr 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743

This theorem is referenced by:  preq12i  3805  preq12d  3808  preq12b  4128  ce2  6193