Theorem pm2.21i 655

Description: A contradiction implies anything. Inference from pm2.21 626. (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)

Hypothesis

Ref	Expression
pm2.21i.1	|- -. ph

Assertion

Ref	Expression
pm2.21i	|- ( ph -> ps )

Proof of Theorem pm2.21i

Step	Hyp	Ref	Expression
1		pm2.21i.1	. 2 |- -. ph
2		pm2.21 626	. 2 |- ( -. ph -> ( ph -> ps ) )
3	1, 2	ax-mp 5	1 |- ( ph -> ps )

This proof depends on syntax axioms:   -. wn 3   -> wi 4

This proof depends on axioms:  ax-mp 5  ax-in2 624

This theorem is used by:  pm2.24ii  656  2false  713  pm3.2ni  825  falim  1416  pclem6  1423  dcfromcon  1498  nfnth  1518  alnex  1552  ax4sp1  1586  rex0  3539  0ss  3561  abf  3570  ral0  3629  rabsnifsb  3777  int0  3984  relndmfv  5728  nnsucelsuc  6764  nnmordi  6789  nnaordex  6801  0er  6841  fiintim  7238  indval0  9300  elnnnn0b  9612  xltnegi  10248  xnn0xadd0  10280  frec2uzltd  10855  hashf1lem2  11302  sum0  12174  fsum2dlemstep  12220  prod0  12371  fprod2dlemstep  12408  nn0enne  12688  exprmfct  12936  prm23lt5  13065  4sqlem18  13210  prmlem1a  13244  prmlem2  13257  0met  15576  ppiublem1  16252  ppiublem2  16253  lgsdir2lem3  16315  gausslemma2dlem0i  16342  2lgs  16389  2lgsoddprmlem3  16396  vtxdg0v  16701  clwwlkn0  16815  clwwlk0on0  16838