Theorem preq12d 3608

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 3602	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷})
4	1, 2, 3	syl2anc 408	1 ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷})

Syntax hints:   → wi 4   = wceq 1331  {cpr 3528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534

This theorem is referenced by:  opeq1  3705  opeq2  3706  xrminrecl  11042  xrminadd  11044  xmetxp  12676  xmetxpbl  12677  txmetcnp  12687