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Theorem hb3and 1393
 Description: Deduction form of bound-variable hypothesis builder hb3an 1456. (Contributed by NM, 17-Feb-2013.)
Hypotheses
Ref Expression
hb3and.1 (𝜑 → (𝜓 → ∀𝑥𝜓))
hb3and.2 (𝜑 → (𝜒 → ∀𝑥𝜒))
hb3and.3 (𝜑 → (𝜃 → ∀𝑥𝜃))
Assertion
Ref Expression
hb3and (𝜑 → ((𝜓𝜒𝜃) → ∀𝑥(𝜓𝜒𝜃)))

Proof of Theorem hb3and
StepHypRef Expression
1 hb3and.1 . . 3 (𝜑 → (𝜓 → ∀𝑥𝜓))
2 hb3and.2 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
3 hb3and.3 . . 3 (𝜑 → (𝜃 → ∀𝑥𝜃))
41, 2, 33anim123d 1223 . 2 (𝜑 → ((𝜓𝜒𝜃) → (∀𝑥𝜓 ∧ ∀𝑥𝜒 ∧ ∀𝑥𝜃)))
5 19.26-3an 1386 . 2 (∀𝑥(𝜓𝜒𝜃) ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒 ∧ ∀𝑥𝜃))
64, 5syl6ibr 155 1 (𝜑 → ((𝜓𝜒𝜃) → ∀𝑥(𝜓𝜒𝜃)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ w3a 894  ∀wal 1255 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1350  ax-gen 1352 This theorem depends on definitions:  df-bi 114  df-3an 896 This theorem is referenced by: (None)
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