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Theorem bnj23 30038
 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 3176 . . . . 5 𝑤 ∈ V
2 sbcng 3443 . . . . 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 2680 . . . . . . 7 (𝑤𝐵𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑})
6 nfcv 2751 . . . . . . . 8 𝑥𝐴
76elrabsf 3441 . . . . . . 7 (𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑤𝐴[𝑤 / 𝑥] ¬ 𝜑))
85, 7bitri 263 . . . . . 6 (𝑤𝐵 ↔ (𝑤𝐴[𝑤 / 𝑥] ¬ 𝜑))
9 breq1 4586 . . . . . . . 8 (𝑧 = 𝑤 → (𝑧𝑅𝑦𝑤𝑅𝑦))
109notbid 307 . . . . . . 7 (𝑧 = 𝑤 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑤𝑅𝑦))
1110rspccv 3279 . . . . . 6 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → (𝑤𝐵 → ¬ 𝑤𝑅𝑦))
128, 11syl5bir 232 . . . . 5 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → ((𝑤𝐴[𝑤 / 𝑥] ¬ 𝜑) → ¬ 𝑤𝑅𝑦))
1312expdimp 452 . . . 4 ((∀𝑧𝐵 ¬ 𝑧𝑅𝑦𝑤𝐴) → ([𝑤 / 𝑥] ¬ 𝜑 → ¬ 𝑤𝑅𝑦))
143, 13syl5bir 232 . . 3 ((∀𝑧𝐵 ¬ 𝑧𝑅𝑦𝑤𝐴) → (¬ [𝑤 / 𝑥]𝜑 → ¬ 𝑤𝑅𝑦))
1514con4d 113 . 2 ((∀𝑧𝐵 ¬ 𝑧𝑅𝑦𝑤𝐴) → (𝑤𝑅𝑦[𝑤 / 𝑥]𝜑))
1615ralrimiva 2949 1 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤𝐴 (𝑤𝑅𝑦[𝑤 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  [wsbc 3402   class class class wbr 4583 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-rab 2905  df-v 3175  df-sbc 3403  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 4584 This theorem is referenced by:  bnj110  30182
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