Theorem sylan 59

Description: Syllogism inference. (Contributed by Mario Carneiro, 8-Oct-2014.)

Hypotheses

Ref	Expression
sylan.1	|- R |= S
sylan.2	|- (S, T) |= A

Assertion

Ref	Expression
sylan	|- (R, T) |= A

Proof of Theorem sylan

Step	Hyp	Ref	Expression
1		sylan.1	. . 3 |- R |= S
2		sylan.2	. . . . 5 |- (S, T) |= A
3	2	ax-cb1 29	. . . 4 |- (S, T):*
4	3	wctr 34	. . 3 |- T:*
5	1, 4	adantr 55	. 2 |- (R, T) |= S
6	1	ax-cb1 29	. . 3 |- R:*
7	6, 4	simpr 23	. 2 |- (R, T) |= T
8	5, 7, 2	syl2anc 19	1 |- (R, T) |= A

Syntax hints:  kct 10   |= wffMMJ2 11

This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-cb1 29  ax-wctr 32

This theorem is referenced by:  anasss  61