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	⊢ ¬ 𝜑

Assertion

Ref	Expression
pm2.21i	⊢ (𝜑 → 𝜓)

Proof of Theorem pm2.21i

Step	Hyp	Ref	Expression
1		pm2.21i.1	. 2 ⊢ ¬ 𝜑
2		pm2.21 620	. 2 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
3	1, 2	ax-mp 5	1 ⊢ (𝜑 → 𝜓)

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  3937  nnsucelsuc  6645  nnmordi  6670  nnaordex  6682  0er  6722  fiintim  7101  elnnnn0b  9421  xltnegi  10039  xnn0xadd0  10071  frec2uzltd  10633  sum0  11907  fsum2dlemstep  11953  prod0  12104  fprod2dlemstep  12141  nn0enne  12421  exprmfct  12668  prm23lt5  12794  4sqlem18  12939  0met  15066  lgsdir2lem3  15717  gausslemma2dlem0i  15744  2lgs  15791  2lgsoddprmlem3  15798