Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ad5ant235 Structured version   Visualization version   GIF version

 Description: Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
Hypothesis
Ref Expression
ad5ant235.1 ((𝜑𝜓𝜒) → 𝜃)
Assertion
Ref Expression
ad5ant235 (((((𝜏𝜑) ∧ 𝜓) ∧ 𝜂) ∧ 𝜒) → 𝜃)

Proof of Theorem ad5ant235
StepHypRef Expression
1 ad5ant235.1 . . . . . . . 8 ((𝜑𝜓𝜒) → 𝜃)
213exp 1256 . . . . . . 7 (𝜑 → (𝜓 → (𝜒𝜃)))
32a1ddd 78 . . . . . 6 (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃))))
43a1ddd 78 . . . . 5 (𝜑 → (𝜓 → (𝜒 → (𝜂 → (𝜏𝜃)))))
54com5r 102 . . . 4 (𝜏 → (𝜑 → (𝜓 → (𝜒 → (𝜂𝜃)))))
65com45 95 . . 3 (𝜏 → (𝜑 → (𝜓 → (𝜂 → (𝜒𝜃)))))
76imp 444 . 2 ((𝜏𝜑) → (𝜓 → (𝜂 → (𝜒𝜃))))
87imp41 617 1 (((((𝜏𝜑) ∧ 𝜓) ∧ 𝜂) ∧ 𝜒) → 𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 196  df-an 385  df-3an 1033 This theorem is referenced by: (None)
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