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  1753  rb-ax3  1754  rb-ax4  1755  spimfw  1965  hba1w  2048  hbe1a  2145  sp  2184  axc4  2320  exmoeu  2574  necon1bi  2953  fvrn0  6854  nfunsn  6866  mpoxneldm  8152  mpoxopxnop0  8155  ixpprc  8853  fineqv  9168  unbndrank  9757  pf1rcl  22252  stri  32219  hstri  32227  ddemeas  34202  hbntg  35778  bj-modalb  36689  hba1-o  38875  axc5c711  38896  naecoms-o  38905  axc5c4c711  44374  hbntal  44527  resinsnlem  48856