Theorem ax1 203

Description: Axiom Simp. Axiom A1 of [Margaris] p. 49. (Contributed by Mario Carneiro, 9-Oct-2014.)

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 43	. . . . 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 55	. . 3 |- ((T., R), S) |= R
6	5	ex 158	. 2 |- (T., R) |= [S ==> R]
7	6	ex 158	1 |- T. |= [R ==> [S ==> R]]

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