Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1185 Structured version   Visualization version   GIF version

Theorem bnj1185 33804
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 5152 . . . . . 6 (𝑤 = 𝑦 → (𝑤𝑅𝑧𝑦𝑅𝑧))
32notbid 318 . . . . 5 (𝑤 = 𝑦 → (¬ 𝑤𝑅𝑧 ↔ ¬ 𝑦𝑅𝑧))
43cbvralvw 3235 . . . 4 (∀𝑤𝐵 ¬ 𝑤𝑅𝑧 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧)
54rexbii 3095 . . 3 (∃𝑧𝐵𝑤𝐵 ¬ 𝑤𝑅𝑧 ↔ ∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧)
61, 5sylib 217 . 2 (𝜑 → ∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧)
7 eleq1w 2817 . . . . 5 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
8 breq2 5153 . . . . . . 7 (𝑧 = 𝑥 → (𝑦𝑅𝑧𝑦𝑅𝑥))
98notbid 318 . . . . . 6 (𝑧 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑦𝑅𝑥))
109ralbidv 3178 . . . . 5 (𝑧 = 𝑥 → (∀𝑦𝐵 ¬ 𝑦𝑅𝑧 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
117, 10anbi12d 632 . . . 4 (𝑧 = 𝑥 → ((𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧) ↔ (𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)))
1211cbvexvw 2041 . . 3 (∃𝑧(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧) ↔ ∃𝑥(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
13 df-rex 3072 . . 3 (∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧 ↔ ∃𝑧(𝑧𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑧))
14 df-rex 3072 . . 3 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥(𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
1512, 13, 143bitr4ri 304 . 2 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑧𝐵𝑦𝐵 ¬ 𝑦𝑅𝑧)
166, 15sylibr 233 1 (𝜑 → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wex 1782  wcel 2107  wral 3062  wrex 3071   class class class wbr 5149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150
This theorem is referenced by:  bnj1190  34019  bnj1189  34020
  Copyright terms: Public domain W3C validator