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  6563  unfidisj  6984  fidcenumlemrks  7020  difinfsn  7167  pw1nel3  7300  addnqprlemfl  7628  addnqprlemfu  7629  mulnqprlemfl  7644  mulnqprlemfu  7645  cauappcvgprlemladdru  7725  caucvgprprlemaddq  7777  fzpreddisj  10148  fprodntrivap  11751  pw2dvdslemn  12343  isnsgrp  13059  ivthinclemdisj  14886  dvply1  15011  lgseisenlem1  15321  lgsquadlem3  15330  pwtrufal  15652  pw1nct  15657  nninfsellemsuc  15666