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Theorem bnj23 31887
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj23.1 𝐵 = {𝑥𝐴 ∣ ¬ 𝜑}
Assertion
Ref Expression
bnj23 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤𝐴 (𝑤𝑅𝑦[𝑤 / 𝑥]𝜑))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴,𝑧   𝑤,𝐵,𝑦,𝑧   𝑤,𝑅,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝐴(𝑤)   𝐵(𝑥)   𝑅(𝑥)

Proof of Theorem bnj23
StepHypRef Expression
1 sbcng 3816 . . . . 5 (𝑤 ∈ V → ([𝑤 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑤 / 𝑥]𝜑))
21elv 3497 . . . 4 ([𝑤 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑤 / 𝑥]𝜑)
3 bnj23.1 . . . . . . . 8 𝐵 = {𝑥𝐴 ∣ ¬ 𝜑}
43eleq2i 2901 . . . . . . 7 (𝑤𝐵𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑})
5 nfcv 2974 . . . . . . . 8 𝑥𝐴
65elrabsf 3813 . . . . . . 7 (𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑤𝐴[𝑤 / 𝑥] ¬ 𝜑))
74, 6bitri 276 . . . . . 6 (𝑤𝐵 ↔ (𝑤𝐴[𝑤 / 𝑥] ¬ 𝜑))
8 breq1 5060 . . . . . . . 8 (𝑧 = 𝑤 → (𝑧𝑅𝑦𝑤𝑅𝑦))
98notbid 319 . . . . . . 7 (𝑧 = 𝑤 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑤𝑅𝑦))
109rspccv 3617 . . . . . 6 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → (𝑤𝐵 → ¬ 𝑤𝑅𝑦))
117, 10syl5bir 244 . . . . 5 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → ((𝑤𝐴[𝑤 / 𝑥] ¬ 𝜑) → ¬ 𝑤𝑅𝑦))
1211expdimp 453 . . . 4 ((∀𝑧𝐵 ¬ 𝑧𝑅𝑦𝑤𝐴) → ([𝑤 / 𝑥] ¬ 𝜑 → ¬ 𝑤𝑅𝑦))
132, 12syl5bir 244 . . 3 ((∀𝑧𝐵 ¬ 𝑧𝑅𝑦𝑤𝐴) → (¬ [𝑤 / 𝑥]𝜑 → ¬ 𝑤𝑅𝑦))
1413con4d 115 . 2 ((∀𝑧𝐵 ¬ 𝑧𝑅𝑦𝑤𝐴) → (𝑤𝑅𝑦[𝑤 / 𝑥]𝜑))
1514ralrimiva 3179 1 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤𝐴 (𝑤𝑅𝑦[𝑤 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  [wsbc 3769   class class class wbr 5057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058
This theorem is referenced by:  bnj110  32029
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