Theorem axmp 193

Description: Rule of Modus Ponens. The postulated inference rule of propositional calculus. See e.g. Rule 1 of [Hamilton] p. 73.

Hypotheses

Ref	Expression
axmp.1	⊢ S:∗
axmp.2	⊢ ⊤⊧R
axmp.3	⊢ ⊤⊧[R ⇒ S]

Assertion

Ref	Expression
axmp	⊢ ⊤⊧S

Proof of Theorem axmp

Step	Hyp	Ref	Expression
1		axmp.1	. 2 ⊢ S:∗
2		axmp.2	. 2 ⊢ ⊤⊧R
3		axmp.3	. 2 ⊢ ⊤⊧[R ⇒ S]
4	1, 2, 3	mpd 146	1 ⊢ ⊤⊧S

Syntax hints:  ∗hb 3  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12   ⇒ tim 111

This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103

This theorem depends on definitions:  df-ov 65  df-an 118  df-im 119

This theorem is referenced by: (None)