Theorem mt2 649

Description: A rule similar to modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.)

Hypotheses

Ref	Expression
mt2.1	|- ps
mt2.2	|- ( ph -> -. ps )

Assertion

Ref	Expression
mt2	|- -. ph

Proof of Theorem mt2

Step	Hyp	Ref	Expression
1		mt2.1	. . 3 |- ps
2	1	a1i 9	. 2 |- ( ph -> ps )
3		mt2.2	. 2 |- ( ph -> -. ps )
4	2, 3	pm2.65i 648	1 |- -. ph

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:  pw1ne3  7589  3nelsucpw1  7593  3nsssucpw1  7595  0nnn  9331  nn0ge2m1nn  9627  xsubge0  10283  xnn0nnen  10874  konigsberg  16734  3dom  17018