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  3509  0ss  3530  abf  3535  ral0  3593  int0  3936  nnsucelsuc  6635  nnmordi  6660  nnaordex  6672  0er  6712  fiintim  7089  elnnnn0b  9409  xltnegi  10027  xnn0xadd0  10059  frec2uzltd  10620  sum0  11894  fsum2dlemstep  11940  prod0  12091  fprod2dlemstep  12128  nn0enne  12408  exprmfct  12655  prm23lt5  12781  4sqlem18  12926  0met  15052  lgsdir2lem3  15703  gausslemma2dlem0i  15730  2lgs  15777  2lgsoddprmlem3  15784