Theorem mt2d 634

Description: Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)

Hypotheses

Ref	Expression
mt2d.1	⊢ (𝜑 → 𝜒)
mt2d.2	⊢ (𝜑 → (𝜓 → ¬ 𝜒))

Assertion

Ref	Expression
mt2d	⊢ (𝜑 → ¬ 𝜓)

Proof of Theorem mt2d

Step	Hyp	Ref	Expression
1		mt2d.1	. 2 ⊢ (𝜑 → 𝜒)
2		mt2d.2	. . 3 ⊢ (𝜑 → (𝜓 → ¬ 𝜒))
3	2	con2d 633	. 2 ⊢ (𝜑 → (𝜒 → ¬ 𝜓))
4	1, 3	mpd 13	1 ⊢ (𝜑 → ¬ 𝜓)

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-in1 623  ax-in2 624

This theorem is used by:  nsyl3  635  mt2i  653  en2lp  4703  recnz  9751  xnn0dcle  10222  fznuz  10527  uznfz  10528  nninfctlemfo  12843  pcadd  13149  ballotfilemfrcn0  13332  oddennn  13342  mvrf1fi  15215  perfectlem1  16311  lgseisenlem1  16406