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Theorem bnj1171 31400
Description: Technical lemma for bnj69 31410. 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
bnj1171.13 ((𝜑𝜓) → 𝐵𝐴)
bnj1171.129 𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
Assertion
Ref Expression
bnj1171 𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧)))

Proof of Theorem bnj1171
StepHypRef Expression
1 bnj1171.129 . 2 𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
2 bnj1171.13 . . . . . . . . . . 11 ((𝜑𝜓) → 𝐵𝐴)
32sseld 3749 . . . . . . . . . 10 ((𝜑𝜓) → (𝑤𝐵𝑤𝐴))
43pm4.71rd 544 . . . . . . . . 9 ((𝜑𝜓) → (𝑤𝐵 ↔ (𝑤𝐴𝑤𝐵)))
54imbi1d 330 . . . . . . . 8 ((𝜑𝜓) → ((𝑤𝐵 → ¬ 𝑤𝑅𝑧) ↔ ((𝑤𝐴𝑤𝐵) → ¬ 𝑤𝑅𝑧)))
6 impexp 437 . . . . . . . 8 (((𝑤𝐴𝑤𝐵) → ¬ 𝑤𝑅𝑧) ↔ (𝑤𝐴 → (𝑤𝐵 → ¬ 𝑤𝑅𝑧)))
75, 6syl6bb 276 . . . . . . 7 ((𝜑𝜓) → ((𝑤𝐵 → ¬ 𝑤𝑅𝑧) ↔ (𝑤𝐴 → (𝑤𝐵 → ¬ 𝑤𝑅𝑧))))
8 con2b 348 . . . . . . . 8 ((𝑤𝑅𝑧 → ¬ 𝑤𝐵) ↔ (𝑤𝐵 → ¬ 𝑤𝑅𝑧))
98imbi2i 325 . . . . . . 7 ((𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)) ↔ (𝑤𝐴 → (𝑤𝐵 → ¬ 𝑤𝑅𝑧)))
107, 9syl6bbr 278 . . . . . 6 ((𝜑𝜓) → ((𝑤𝐵 → ¬ 𝑤𝑅𝑧) ↔ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
1110anbi2d 606 . . . . 5 ((𝜑𝜓) → ((𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧)) ↔ (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
1211pm5.74i 260 . . . 4 (((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧))) ↔ ((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
1312albii 1894 . . 3 (∀𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧))) ↔ ∀𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
1413exbii 1923 . 2 (∃𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧))) ↔ ∃𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
151, 14mpbir 221 1 𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wal 1628  wex 1851  wcel 2144  wss 3721   class class class wbr 4784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-in 3728  df-ss 3735
This theorem is referenced by:  bnj1190  31408
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