Theorem ax1 190

Description: Axiom Simp. Axiom A1 of [Margaris] p. 49.

Hypotheses

Ref	Expression
ax1.1	|- R:*
ax1.2	|- S:*

Assertion

Ref	Expression
ax1	|- T. |= [R ==> [S ==> R]]

Proof of Theorem ax1

Step	Hyp	Ref	Expression
1		wtru 40	. . . . 5 |- T.:*
2		ax1.1	. . . . 5 |- R:*
3	1, 2	simpr 23	. . . 4 |- (T., R) |= R
4		ax1.2	. . . 4 |- S:*
5	3, 4	adantr 50	. . 3 |- ((T., R), S) |= R
6	5	ex 148	. 2 |- (T., R) |= [S ==> R]
7	6	ex 148	1 |- T. |= [R ==> [S ==> R]]

Syntax hints:  *hb 3  T.kt 8  [kbr 9  kct 10   |= 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)