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Theorem bnj1049 32254
Description: Technical lemma for bnj69 32290. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1049.1 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
bnj1049.2 (𝜂 ↔ ((𝜃𝜏𝜒𝜁) → 𝑧𝐵))
Assertion
Ref Expression
bnj1049 (∀𝑖𝑛 𝜂 ↔ ∀𝑖𝜂)

Proof of Theorem bnj1049
StepHypRef Expression
1 df-ral 3131 . 2 (∀𝑖𝑛 𝜂 ↔ ∀𝑖(𝑖𝑛𝜂))
2 bnj1049.2 . . . . . . 7 (𝜂 ↔ ((𝜃𝜏𝜒𝜁) → 𝑧𝐵))
32imbi2i 339 . . . . . 6 ((𝑖𝑛𝜂) ↔ (𝑖𝑛 → ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)))
4 impexp 454 . . . . . 6 (((𝑖𝑛 ∧ (𝜃𝜏𝜒𝜁)) → 𝑧𝐵) ↔ (𝑖𝑛 → ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)))
53, 4bitr4i 281 . . . . 5 ((𝑖𝑛𝜂) ↔ ((𝑖𝑛 ∧ (𝜃𝜏𝜒𝜁)) → 𝑧𝐵))
6 bnj1049.1 . . . . . . . . . 10 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
76simplbi 501 . . . . . . . . 9 (𝜁𝑖𝑛)
87bnj708 32035 . . . . . . . 8 ((𝜃𝜏𝜒𝜁) → 𝑖𝑛)
98pm4.71ri 564 . . . . . . 7 ((𝜃𝜏𝜒𝜁) ↔ (𝑖𝑛 ∧ (𝜃𝜏𝜒𝜁)))
109bicomi 227 . . . . . 6 ((𝑖𝑛 ∧ (𝜃𝜏𝜒𝜁)) ↔ (𝜃𝜏𝜒𝜁))
1110imbi1i 353 . . . . 5 (((𝑖𝑛 ∧ (𝜃𝜏𝜒𝜁)) → 𝑧𝐵) ↔ ((𝜃𝜏𝜒𝜁) → 𝑧𝐵))
125, 11bitri 278 . . . 4 ((𝑖𝑛𝜂) ↔ ((𝜃𝜏𝜒𝜁) → 𝑧𝐵))
1312, 2bitr4i 281 . . 3 ((𝑖𝑛𝜂) ↔ 𝜂)
1413albii 1821 . 2 (∀𝑖(𝑖𝑛𝜂) ↔ ∀𝑖𝜂)
151, 14bitri 278 1 (∀𝑖𝑛 𝜂 ↔ ∀𝑖𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wcel 2115  wral 3126  cfv 6328  w-bnj17 31964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 210  df-an 400  df-ral 3131  df-bnj17 31965
This theorem is referenced by:  bnj1052  32255
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