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Theorem bnj1185 29912
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1185.1 (𝜑 → ∃𝑧𝐵𝑤𝐵 ¬ 𝑤𝑅𝑧)
Assertion
Ref Expression
bnj1185 (𝜑 → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Distinct variable groups:   𝑤,𝐵,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑤,𝑅,𝑦,𝑧   𝑥,𝑅
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem bnj1185
StepHypRef Expression
1 bnj1185.1 . . 3 (𝜑 → ∃𝑧𝐵𝑤𝐵 ¬ 𝑤𝑅𝑧)
2 breq1 4581 . . . . . 6 (𝑤 = 𝑦 → (𝑤𝑅𝑧𝑦𝑅𝑧))
32notbid 307 . . . . 5 (𝑤 = 𝑦 → (¬ 𝑤𝑅𝑧 ↔ ¬ 𝑦𝑅𝑧))
43cbvralv 3147 . . . 4 (∀𝑤𝐵 ¬ 𝑤𝑅𝑧 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧)
54rexbii 3023 . . 3 (∃𝑧𝐵𝑤𝐵 ¬ 𝑤𝑅𝑧 ↔ ∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧)
61, 5sylib 207 . 2 (𝜑 → ∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧)
7 eleq1 2676 . . . . 5 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
8 breq2 4582 . . . . . . 7 (𝑧 = 𝑥 → (𝑦𝑅𝑧𝑦𝑅𝑥))
98notbid 307 . . . . . 6 (𝑧 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑦𝑅𝑥))
109ralbidv 2969 . . . . 5 (𝑧 = 𝑥 → (∀𝑦𝐵 ¬ 𝑦𝑅𝑧 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
117, 10anbi12d 743 . . . 4 (𝑧 = 𝑥 → ((𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧) ↔ (𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)))
1211cbvexv 2263 . . 3 (∃𝑧(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧) ↔ ∃𝑥(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
13 df-rex 2902 . . 3 (∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧 ↔ ∃𝑧(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧))
14 df-rex 2902 . . 3 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
1512, 13, 143bitr4ri 292 . 2 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧)
166, 15sylibr 223 1 (𝜑 → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wex 1695  wcel 1977  wral 2896  wrex 2897   class class class wbr 4578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4579
This theorem is referenced by:  bnj1190  30124  bnj1189  30125
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