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Theorem nincom 3226
Description: Anti-intersection commutes. (Contributed by SF, 10-Jan-2015.)
Assertion
Ref Expression
nincom (AB) = (BA)

Proof of Theorem nincom
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 nancom 1290 . . 3 ((x A x B) ↔ (x B x A))
2 vex 2862 . . . 4 x V
32elnin 3224 . . 3 (x (AB) ↔ (x A x B))
42elnin 3224 . . 3 (x (BA) ↔ (x B x A))
51, 3, 43bitr4i 268 . 2 (x (AB) ↔ x (BA))
65eqriv 2350 1 (AB) = (BA)
Colors of variables: wff setvar class
Syntax hints:   wnan 1287   = wceq 1642   wcel 1710  cnin 3204
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 2478  df-v 2861  df-nin 3211
This theorem is referenced by:  nineq2  3235
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