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  887  pm3.13  996  rb-ax2  1754  rb-ax3  1755  rb-ax4  1756  spimfw  1966  hba1w  2050  hbe1a  2149  sp  2188  axc4  2324  exmoeu  2578  necon1bi  2957  fvrn0  6856  nfunsn  6867  mpoxneldm  8148  mpoxopxnop0  8151  ixpprc  8849  fineqv  9158  unbndrank  9742  pf1rcl  22265  stri  32239  hstri  32247  ddemeas  34270  hbntg  35868  bj-modalb  36781  hba1-o  39016  axc5c711  39037  naecoms-o  39046  axc5c4c711  44518  hbntal  44670  resinsnlem  48995