Theorem preq12i 3704

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

Hypotheses

Ref	Expression
preq1i.1	⊢ 𝐴 = 𝐵
preq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
preq12i	⊢ {𝐴, 𝐶} = {𝐵, 𝐷}

Proof of Theorem preq12i

Step	Hyp	Ref	Expression
1		preq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		preq12i.2	. 2 ⊢ 𝐶 = 𝐷
3		preq12 3701	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷})
4	1, 2, 3	mp2an 426	1 ⊢ {𝐴, 𝐶} = {𝐵, 𝐷}

Syntax hints:   = wceq 1364  {cpr 3623

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629

This theorem is referenced by:  lgsdir2lem5  15273