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Theorem mpsylsyld 69
Description: Modus ponens combined with a double syllogism inference. (Contributed by Alan Sare, 22-Jul-2012.)
Hypotheses
Ref Expression
mpsylsyld.1 𝜑
mpsylsyld.2 (𝜓 → (𝜒𝜃))
mpsylsyld.3 (𝜑 → (𝜃𝜏))
Assertion
Ref Expression
mpsylsyld (𝜓 → (𝜒𝜏))

Proof of Theorem mpsylsyld
StepHypRef Expression
1 mpsylsyld.1 . . 3 𝜑
21a1i 11 . 2 (𝜓𝜑)
3 mpsylsyld.2 . 2 (𝜓 → (𝜒𝜃))
4 mpsylsyld.3 . 2 (𝜑 → (𝜃𝜏))
52, 3, 4sylsyld 61 1 (𝜓 → (𝜒𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7
This theorem is referenced by:  r1sdom  9202  r1ord3g  9207  r1ord2  9209  rlimclim  14905  vk15.4j  41182  onfrALTlem3  41198  ee02an  41353
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