Theorem sylnibr 678

Description: A mixed syllogism inference from an implication and a biconditional. Useful for substituting an consequent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.)

Hypotheses

Ref	Expression
sylnibr.1	⊢ (𝜑 → ¬ 𝜓)
sylnibr.2	⊢ (𝜒 ↔ 𝜓)

Assertion

Ref	Expression
sylnibr	⊢ (𝜑 → ¬ 𝜒)

Proof of Theorem sylnibr

Step	Hyp	Ref	Expression
1		sylnibr.1	. 2 ⊢ (𝜑 → ¬ 𝜓)
2		sylnibr.2	. . 3 ⊢ (𝜒 ↔ 𝜓)
3	2	bicomi 132	. 2 ⊢ (𝜓 ↔ 𝜒)
4	1, 3	sylnib 677	1 ⊢ (𝜑 → ¬ 𝜒)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  rexnalim  2486  nssr  3244  difdif  3289  unssin  3403  inssun  3404  undif3ss  3425  ssdif0im  3516  dcun  3561  prneimg  3805  iundif2ss  3983  nssssr  4256  pofun  4348  frirrg  4386  regexmidlem1  4570  dcdifsnid  6571  unfidisj  6992  fidcenumlemrks  7028  difinfsn  7175  pw1nel3  7314  addnqprlemfl  7643  addnqprlemfu  7644  mulnqprlemfl  7659  mulnqprlemfu  7660  cauappcvgprlemladdru  7740  caucvgprprlemaddq  7792  fzpreddisj  10163  fprodntrivap  11766  pw2dvdslemn  12358  isnsgrp  13108  ivthinclemdisj  14960  dvply1  15085  lgseisenlem1  15395  lgsquadlem3  15404  pwtrufal  15728  pw1nct  15734  nninfsellemsuc  15743