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Theorem hband 1394
Description: Deduction form of bound-variable hypothesis builder hban 1455. (Contributed by NM, 2-Jan-2002.)
Hypotheses
Ref Expression
hband.1 (𝜑 → (𝜓 → ∀𝑥𝜓))
hband.2 (𝜑 → (𝜒 → ∀𝑥𝜒))
Assertion
Ref Expression
hband (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))

Proof of Theorem hband
StepHypRef Expression
1 hband.1 . . 3 (𝜑 → (𝜓 → ∀𝑥𝜓))
2 hband.2 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
31, 2anim12d 322 . 2 (𝜑 → ((𝜓𝜒) → (∀𝑥𝜓 ∧ ∀𝑥𝜒)))
4 19.26 1386 . 2 (∀𝑥(𝜓𝜒) ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))
53, 4syl6ibr 155 1 (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257
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 1352  ax-gen 1354
This theorem depends on definitions:  df-bi 114
This theorem is referenced by: (None)
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