Theorem con1i 148

Description: A contraposition inference. Inference associated with con1 147. Its associated inference is mt3 204. (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 23	. 2 ⊢ (¬ 𝜓 → ¬ 𝜓)
2		con1i.1	. 2 ⊢ (¬ 𝜑 → 𝜓)
3	1, 2	nsyl2 142	1 ⊢ (¬ 𝜓 → 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by:  pm2.24i  151  nsyl4  159  nsyl5  160  impi  165  simplim  168  nbior  901  pm3.13  1010  rb-ax2  1786  rb-ax3  1787  rb-ax4  1788  spimfw  1998  hba1w  2082  hbe1a  2181  sp  2220  axc4  2352  exmoeu  2607  necon1bi  2984  fvrn0  6905  nfunsn  6916  mpoxneldm  8213  mpoxopxnop0  8216  ixpprc  8931  fineqv  9242  unbndrank  9836  pf1rcl  22647  stri  32841  hstri  32849  ddemeas  34851  hbntg  36537  bj-modalb  37590  hba1-o  39922  axc5c711  39943  naecoms-o  39952  axc5c4c711  45344  hbntal  45495  resinsnlem  49923