Theorem preq1 3800

Description: Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)

Assertion

Ref	Expression
preq1	⊢ (A = B → {A, C} = {B, C})

Proof of Theorem preq1

Step	Hyp	Ref	Expression
1		sneq 3745	. . 3 ⊢ (A = B → {A} = {B})
2	1	uneq1d 3418	. 2 ⊢ (A = B → ({A} ∪ {C}) = ({B} ∪ {C}))
3		df-pr 3743	. 2 ⊢ {A, C} = ({A} ∪ {C})
4		df-pr 3743	. 2 ⊢ {B, C} = ({B} ∪ {C})
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → {A, C} = {B, C})

Syntax hints:   → wi 4   = wceq 1642   ∪ cun 3208  {csn 3738  {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:  preq2  3801  preq12  3802  preq1i  3803  preq1d  3806  tpeq1  3809  prnzg  3837  uniprg  3907  intprg  3961  opkeq1  4060  preq12b  4128  preq12bg  4129  enprmapc  6084