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  2322  exmoeu  2581  necon1bi  2961  fvrn0  6911  nfunsn  6923  mpoxneldm  8216  mpoxopxnop0  8219  ixpprc  8938  fineqv  9276  unbndrank  9861  pf1rcl  22292  stri  32243  hstri  32251  ddemeas  34272  hbntg  35828  bj-modalb  36739  hba1-o  38920  axc5c711  38941  naecoms-o  38950  axc5c4c711  44392  hbntal  44545  resinsnlem  48813