Theorem preq12d 3719

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 3713	. 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 1373  {cpr 3635

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3171  df-sn 3640  df-pr 3641

This theorem is referenced by:  opeq1  3821  opeq2  3822  xrminrecl  11628  xrminadd  11630  prdsval  13149  xpsfval  13224  xpsval  13228  ring1  13865  xmetxp  15023  xmetxpbl  15024  txmetcnp  15034  hovera  15163  hoverb  15164  hoverlt1  15165  hovergt0  15166  ivthdich  15169