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  6769  nnmordi  6794  nnaordex  6806  0er  6846  fiintim  7244  indval0  9306  elnnnn0b  9618  xltnegi  10253  xnn0xadd0  10285  frec2uzltd  10860  hashf1lem2  11307  sum0  12179  fsum2dlemstep  12225  prod0  12376  fprod2dlemstep  12413  nn0enne  12693  exprmfct  12941  prm23lt5  13070  4sqlem18  13215  prmlem1a  13249  prmlem2  13262  0met  15627  ppiublem1  16308  ppiublem2  16309  lgsdir2lem3  16371  gausslemma2dlem0i  16398  2lgs  16445  2lgsoddprmlem3  16452  vtxdg0v  16757  clwwlkn0  16871  clwwlk0on0  16894