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  6888  nfunsn  6900  mpoxneldm  8191  mpoxopxnop0  8194  ixpprc  8892  fineqv  9210  unbndrank  9795  pf1rcl  22236  stri  32186  hstri  32194  ddemeas  34226  hbntg  35793  bj-modalb  36704  hba1-o  38890  axc5c711  38911  naecoms-o  38920  axc5c4c711  44390  hbntal  44543  resinsnlem  48856