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  2147  sp  2186  axc4  2322  exmoeu  2576  necon1bi  2956  fvrn0  6850  nfunsn  6861  mpoxneldm  8142  mpoxopxnop0  8145  ixpprc  8843  fineqv  9151  unbndrank  9732  pf1rcl  22262  stri  32232  hstri  32240  ddemeas  34244  hbntg  35838  bj-modalb  36749  hba1-o  38935  axc5c711  38956  naecoms-o  38965  axc5c4c711  44433  hbntal  44585  resinsnlem  48901