Theorem con1i 147

Description: A contraposition inference. Inference associated with con1 146. Its associated inference is mt3 201. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Jun-2013.)

Hypothesis

Ref	Expression
con1i.1	⊢ (¬ 𝜑 → 𝜓)

Assertion

Ref	Expression
con1i	⊢ (¬ 𝜓 → 𝜑)

Proof of Theorem con1i

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (¬ 𝜓 → ¬ 𝜓)
2		con1i.1	. 2 ⊢ (¬ 𝜑 → 𝜓)
3	1, 2	nsyl2 141	1 ⊢ (¬ 𝜓 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  pm2.24i  150  nsyl4  158  nsyl5  159  impi  164  simplim  167  nbior  888  pm3.13  997  rb-ax2  1755  rb-ax3  1756  rb-ax4  1757  spimfw  1967  hba1w  2051  hbe1a  2150  sp  2191  axc4  2326  exmoeu  2581  necon1bi  2960  fvrn0  6868  nfunsn  6879  mpoxneldm  8162  mpoxopxnop0  8165  ixpprc  8867  fineqv  9177  unbndrank  9766  pf1rcl  22314  stri  32328  hstri  32336  ddemeas  34380  hbntg  35985  bj-modalb  37015  hba1-o  39343  axc5c711  39364  naecoms-o  39373  axc5c4c711  44828  hbntal  44980  resinsnlem  49346