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Theorem bnj1185 32344
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 5033 . . . . . 6 (𝑤 = 𝑦 → (𝑤𝑅𝑧𝑦𝑅𝑧))
32notbid 321 . . . . 5 (𝑤 = 𝑦 → (¬ 𝑤𝑅𝑧 ↔ ¬ 𝑦𝑅𝑧))
43cbvralvw 3349 . . . 4 (∀𝑤𝐵 ¬ 𝑤𝑅𝑧 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧)
54rexbii 3161 . . 3 (∃𝑧𝐵𝑤𝐵 ¬ 𝑤𝑅𝑧 ↔ ∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧)
61, 5sylib 221 . 2 (𝜑 → ∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧)
7 eleq1w 2815 . . . . 5 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
8 breq2 5034 . . . . . . 7 (𝑧 = 𝑥 → (𝑦𝑅𝑧𝑦𝑅𝑥))
98notbid 321 . . . . . 6 (𝑧 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑦𝑅𝑥))
109ralbidv 3109 . . . . 5 (𝑧 = 𝑥 → (∀𝑦𝐵 ¬ 𝑦𝑅𝑧 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
117, 10anbi12d 634 . . . 4 (𝑧 = 𝑥 → ((𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧) ↔ (𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)))
1211cbvexvw 2049 . . 3 (∃𝑧(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧) ↔ ∃𝑥(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
13 df-rex 3059 . . 3 (∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧 ↔ ∃𝑧(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧))
14 df-rex 3059 . . 3 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
1512, 13, 143bitr4ri 307 . 2 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧)
166, 15sylibr 237 1 (𝜑 → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wex 1786  wcel 2114  wral 3053  wrex 3054   class class class wbr 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-un 3848  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031
This theorem is referenced by:  bnj1190  32559  bnj1189  32560
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