Theorem con1i 148

Description: A contraposition inference. Inference associated with con1 147. Its associated inference is mt3 204. (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 23	. 2 ⊢ (¬ 𝜓 → ¬ 𝜓)
2		con1i.1	. 2 ⊢ (¬ 𝜑 → 𝜓)
3	1, 2	nsyl2 142	1 ⊢ (¬ 𝜓 → 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by:  pm2.24i  151  nsyl4  159  nsyl5  160  impi  165  simplim  168  nbior  901  pm3.13  1010  rb-ax2  1786  rb-ax3  1787  rb-ax4  1788  spimfw  1998  hba1w  2082  hbe1a  2182  sp  2222  axc4  2357  exmoeu  2612  necon1bi  2989  fvrn0  6916  nfunsn  6927  mpoxneldm  8217  mpoxopxnop0  8220  ixpprc  8926  fineqv  9237  unbndrank  9824  pf1rcl  22546  stri  32646  hstri  32654  ddemeas  34658  hbntg  36316  bj-modalb  37384  hba1-o  39712  axc5c711  39733  naecoms-o  39742  axc5c4c711  45152  hbntal  45303  resinsnlem  49690