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)

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  151  nsyl4  159  nsyl5  160  impi  165  simplim  168  nbior  900  pm3.13  1010  rb-ax2  1783  rb-ax3  1784  rb-ax4  1785  spimfw  1995  hba1w  2079  hbe1a  2179  sp  2219  axc4  2354  exmoeu  2609  necon1bi  2986  fvrn0  6911  nfunsn  6922  mpoxneldm  8209  mpoxopxnop0  8212  ixpprc  8918  fineqv  9228  unbndrank  9815  pf1rcl  22490  stri  32587  hstri  32595  ddemeas  34604  hbntg  36273  bj-modalb  37321  hba1-o  39649  axc5c711  39670  naecoms-o  39679  axc5c4c711  45091  hbntal  45242  resinsnlem  49626