Theorem axmp 206

Description: Rule of Modus Ponens. The postulated inference rule of propositional calculus. See e.g. Rule 1 of [Hamilton] p. 73. (Contributed by Mario Carneiro, 10-Oct-2014.)

Hypotheses

Ref	Expression
axmp.1	|- S:*
axmp.2	|- T. |= R
axmp.3	|- T. |= [R ==> S]

Assertion

Ref	Expression
axmp	|- T. |= S

Proof of Theorem axmp

Step	Hyp	Ref	Expression
1		axmp.1	. 2 |- S:*
2		axmp.2	. 2 |- T. |= R
3		axmp.3	. 2 |- T. |= [R ==> S]
4	1, 2, 3	mpd 156	1 |- T. |= S

Syntax hints:  *hb 3  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12   ==> tim 121

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-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113

This theorem depends on definitions:  df-ov 73  df-an 128  df-im 129

This theorem is referenced by: (None)