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	⊢ ¬ 𝜑

Assertion

Ref	Expression
pm2.21i	⊢ (𝜑 → 𝜓)

Proof of Theorem pm2.21i

Step	Hyp	Ref	Expression
1		pm2.21i.1	. 2 ⊢ ¬ 𝜑
2		pm2.21 626	. 2 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
3	1, 2	ax-mp 5	1 ⊢ (𝜑 → 𝜓)

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  9298  elnnnn0b  9609  xltnegi  10239  xnn0xadd0  10271  frec2uzltd  10842  hashf1lem2  11288  sum0  12157  fsum2dlemstep  12203  prod0  12354  fprod2dlemstep  12391  nn0enne  12671  exprmfct  12918  prm23lt5  13044  4sqlem18  13189  0met  15487  lgsdir2lem3  16161  gausslemma2dlem0i  16188  2lgs  16235  2lgsoddprmlem3  16242  vtxdg0v  16547  clwwlkn0  16661  clwwlk0on0  16684