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  5731  nnsucelsuc  6768  nnmordi  6793  nnaordex  6805  0er  6845  fiintim  7242  indval0  9306  elnnnn0b  9618  xltnegi  10254  xnn0xadd0  10286  frec2uzltd  10861  hashf1lem2  11308  sum0  12180  fsum2dlemstep  12226  prod0  12377  fprod2dlemstep  12414  nn0enne  12694  exprmfct  12942  prm23lt5  13071  4sqlem18  13216  prmlem1a  13250  prmlem2  13263  0met  15610  ppiublem1  16291  ppiublem2  16292  lgsdir2lem3  16354  gausslemma2dlem0i  16381  2lgs  16428  2lgsoddprmlem3  16435  vtxdg0v  16740  clwwlkn0  16854  clwwlk0on0  16877