Theorem pm3.24 705

Description: Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who call it the "law of contradiction"). (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 2-Feb-2015.)

Assertion

Ref	Expression
pm3.24	|- -. ( ph /\ -. ph )

Proof of Theorem pm3.24

Step	Hyp	Ref	Expression
1		notnot 638	. 2 |- ( ph -> -. -. ph )
2		imnan 701	. 2 |- ( ( ph -> -. -. ph ) <-> -. ( ph /\ -. ph ) )
3	1, 2	mpbi 145	1 |- -. ( ph /\ -. ph )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117

This theorem is used by:  nnexmid  862  pm4.43  962  excxor  1427  nonconne  2432  dfnul4  3522  dfnul3  3524  rabnc  3555  ifeqeqxdc  3687  axnul  4258  fiintim  7238  zeoxor  12636  unennn  13288  lgsquadlem2  16197