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Theorem bnj115 30552
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 2913 . 2 (∀𝑛𝐷 (𝜏𝜃) ↔ ∀𝑛(𝑛𝐷 → (𝜏𝜃)))
3 impexp 462 . . . 4 (((𝑛𝐷𝜏) → 𝜃) ↔ (𝑛𝐷 → (𝜏𝜃)))
43bicomi 214 . . 3 ((𝑛𝐷 → (𝜏𝜃)) ↔ ((𝑛𝐷𝜏) → 𝜃))
54albii 1744 . 2 (∀𝑛(𝑛𝐷 → (𝜏𝜃)) ↔ ∀𝑛((𝑛𝐷𝜏) → 𝜃))
61, 2, 53bitri 286 1 (𝜂 ↔ ∀𝑛((𝑛𝐷𝜏) → 𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wcel 1987  wral 2908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2913
This theorem is referenced by:  bnj953  30770  bnj964  30774  bnj1090  30808  bnj1112  30812
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