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Theorem bnj115 32416
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj115.1 (𝜂 ↔ ∀𝑛𝐷 (𝜏𝜃))
Assertion
Ref Expression
bnj115 (𝜂 ↔ ∀𝑛((𝑛𝐷𝜏) → 𝜃))

Proof of Theorem bnj115
StepHypRef Expression
1 bnj115.1 . 2 (𝜂 ↔ ∀𝑛𝐷 (𝜏𝜃))
2 df-ral 3066 . 2 (∀𝑛𝐷 (𝜏𝜃) ↔ ∀𝑛(𝑛𝐷 → (𝜏𝜃)))
3 impexp 454 . . . 4 (((𝑛𝐷𝜏) → 𝜃) ↔ (𝑛𝐷 → (𝜏𝜃)))
43bicomi 227 . . 3 ((𝑛𝐷 → (𝜏𝜃)) ↔ ((𝑛𝐷𝜏) → 𝜃))
54albii 1827 . 2 (∀𝑛(𝑛𝐷 → (𝜏𝜃)) ↔ ∀𝑛((𝑛𝐷𝜏) → 𝜃))
61, 2, 53bitri 300 1 (𝜂 ↔ ∀𝑛((𝑛𝐷𝜏) → 𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1541  wcel 2110  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817
This theorem depends on definitions:  df-bi 210  df-an 400  df-ral 3066
This theorem is referenced by:  bnj953  32632  bnj964  32636  bnj1090  32672  bnj1112  32676
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