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Theorem sylan 59
Description: Syllogism inference. (Contributed by Mario Carneiro, 8-Oct-2014.)
Hypotheses
Ref Expression
sylan.1 RS
sylan.2 (S, T)⊧A
Assertion
Ref Expression
sylan (R, T)⊧A

Proof of Theorem sylan
StepHypRef Expression
1 sylan.1 . . 3 RS
2 sylan.2 . . . . 5 (S, T)⊧A
32ax-cb1 29 . . . 4 (S, T):∗
43wctr 34 . . 3 T:∗
51, 4adantr 55 . 2 (R, T)⊧S
61ax-cb1 29 . . 3 R:∗
76, 4simpr 23 . 2 (R, T)⊧T
85, 7, 2syl2anc 19 1 (R, T)⊧A
Colors of variables: type var term
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
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