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Theorem bnj23 30912
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 vex 3234 . . . . 5 𝑤 ∈ V
2 sbcng 3509 . . . . 5 (𝑤 ∈ V → ([𝑤 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑤 / 𝑥]𝜑))
31, 2ax-mp 5 . . . 4 ([𝑤 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑤 / 𝑥]𝜑)
4 bnj23.1 . . . . . . . 8 𝐵 = {𝑥𝐴 ∣ ¬ 𝜑}
54eleq2i 2722 . . . . . . 7 (𝑤𝐵𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑})
6 nfcv 2793 . . . . . . . 8 𝑥𝐴
76elrabsf 3507 . . . . . . 7 (𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑤𝐴[𝑤 / 𝑥] ¬ 𝜑))
85, 7bitri 264 . . . . . 6 (𝑤𝐵 ↔ (𝑤𝐴[𝑤 / 𝑥] ¬ 𝜑))
9 breq1 4688 . . . . . . . 8 (𝑧 = 𝑤 → (𝑧𝑅𝑦𝑤𝑅𝑦))
109notbid 307 . . . . . . 7 (𝑧 = 𝑤 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑤𝑅𝑦))
1110rspccv 3337 . . . . . 6 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → (𝑤𝐵 → ¬ 𝑤𝑅𝑦))
128, 11syl5bir 233 . . . . 5 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → ((𝑤𝐴[𝑤 / 𝑥] ¬ 𝜑) → ¬ 𝑤𝑅𝑦))
1312expdimp 452 . . . 4 ((∀𝑧𝐵 ¬ 𝑧𝑅𝑦𝑤𝐴) → ([𝑤 / 𝑥] ¬ 𝜑 → ¬ 𝑤𝑅𝑦))
143, 13syl5bir 233 . . 3 ((∀𝑧𝐵 ¬ 𝑧𝑅𝑦𝑤𝐴) → (¬ [𝑤 / 𝑥]𝜑 → ¬ 𝑤𝑅𝑦))
1514con4d 114 . 2 ((∀𝑧𝐵 ¬ 𝑧𝑅𝑦𝑤𝐴) → (𝑤𝑅𝑦[𝑤 / 𝑥]𝜑))
1615ralrimiva 2995 1 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤𝐴 (𝑤𝑅𝑦[𝑤 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  [wsbc 3468   class class class wbr 4685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686
This theorem is referenced by:  bnj110  31054
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