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Theorem bnj23 31125
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 3354 . . . . 5 𝑤 ∈ V
2 sbcng 3629 . . . . 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 2842 . . . . . . 7 (𝑤𝐵𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑})
6 nfcv 2913 . . . . . . . 8 𝑥𝐴
76elrabsf 3627 . . . . . . 7 (𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑤𝐴[𝑤 / 𝑥] ¬ 𝜑))
85, 7bitri 264 . . . . . 6 (𝑤𝐵 ↔ (𝑤𝐴[𝑤 / 𝑥] ¬ 𝜑))
9 breq1 4790 . . . . . . . 8 (𝑧 = 𝑤 → (𝑧𝑅𝑦𝑤𝑅𝑦))
109notbid 307 . . . . . . 7 (𝑧 = 𝑤 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑤𝑅𝑦))
1110rspccv 3458 . . . . . 6 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → (𝑤𝐵 → ¬ 𝑤𝑅𝑦))
128, 11syl5bir 233 . . . . 5 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → ((𝑤𝐴[𝑤 / 𝑥] ¬ 𝜑) → ¬ 𝑤𝑅𝑦))
1312expdimp 440 . . . 4 ((∀𝑧𝐵 ¬ 𝑧𝑅𝑦𝑤𝐴) → ([𝑤 / 𝑥] ¬ 𝜑 → ¬ 𝑤𝑅𝑦))
143, 13syl5bir 233 . . 3 ((∀𝑧𝐵 ¬ 𝑧𝑅𝑦𝑤𝐴) → (¬ [𝑤 / 𝑥]𝜑 → ¬ 𝑤𝑅𝑦))
1514con4d 115 . 2 ((∀𝑧𝐵 ¬ 𝑧𝑅𝑦𝑤𝐴) → (𝑤𝑅𝑦[𝑤 / 𝑥]𝜑))
1615ralrimiva 3115 1 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤𝐴 (𝑤𝑅𝑦[𝑤 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  [wsbc 3588   class class class wbr 4787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788
This theorem is referenced by:  bnj110  31267
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