Theorem pm2.21i 649

Description: A contradiction implies anything. Inference from pm2.21 620. (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		pm2.21 620	. 2 |- ( -. ph -> ( ph -> ps ) )
3	1, 2	ax-mp 5	1 |- ( ph -> ps )

Syntax hints:   -. wn 3   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-in2 618

This theorem is referenced by:  pm2.24ii  650  2false  706  pm3.2ni  818  falim  1409  pclem6  1416  dcfromcon  1491  nfnth  1511  alnex  1545  ax4sp1  1579  rex0  3510  0ss  3531  abf  3536  ral0  3594  rabsnifsb  3735  int0  3940  nnsucelsuc  6654  nnmordi  6679  nnaordex  6691  0er  6731  fiintim  7116  elnnnn0b  9436  xltnegi  10060  xnn0xadd0  10092  frec2uzltd  10655  sum0  11939  fsum2dlemstep  11985  prod0  12136  fprod2dlemstep  12173  nn0enne  12453  exprmfct  12700  prm23lt5  12826  4sqlem18  12971  0met  15098  lgsdir2lem3  15749  gausslemma2dlem0i  15776  2lgs  15823  2lgsoddprmlem3  15830  vtxdg0v  16100  clwwlkn0  16203  clwwlk0on0  16226