Theorem con1i 121

Description: A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Jun-2013.)

Hypothesis

Ref	Expression
con1i.a	⊢ (¬ φ → ψ)

Assertion

Ref	Expression
con1i	⊢ (¬ ψ → φ)

Proof of Theorem con1i

Step	Hyp	Ref	Expression
1		id 19	. 2 ⊢ (¬ ψ → ¬ ψ)
2		con1i.a	. 2 ⊢ (¬ φ → ψ)
3	1, 2	nsyl2 119	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:  nsyl4  134  pm2.24i  136  impi  140  simplim  143  pm3.13  487  pm5.55  867  rb-ax2  1518  rb-ax3  1519  rb-ax4  1520  ax17e  1618  spimfw  1646  hba1w  1707  ax5o  1749  hbnt  1775  hba1OLD  1787  nfndOLD  1792  hbimdOLD  1816  naecoms  1948  hba1-o  2149  ax467  2169  naecoms-o  2178  necon3bi  2558  necon1ai  2559  nfunsn  5354