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Theorem bnj23 30492
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 3189 . . . . 5 𝑤 ∈ V
2 sbcng 3458 . . . . 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 2690 . . . . . . 7 (𝑤𝐵𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑})
6 nfcv 2761 . . . . . . . 8 𝑥𝐴
76elrabsf 3456 . . . . . . 7 (𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑤𝐴[𝑤 / 𝑥] ¬ 𝜑))
85, 7bitri 264 . . . . . 6 (𝑤𝐵 ↔ (𝑤𝐴[𝑤 / 𝑥] ¬ 𝜑))
9 breq1 4616 . . . . . . . 8 (𝑧 = 𝑤 → (𝑧𝑅𝑦𝑤𝑅𝑦))
109notbid 308 . . . . . . 7 (𝑧 = 𝑤 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑤𝑅𝑦))
1110rspccv 3292 . . . . . 6 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → (𝑤𝐵 → ¬ 𝑤𝑅𝑦))
128, 11syl5bir 233 . . . . 5 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → ((𝑤𝐴[𝑤 / 𝑥] ¬ 𝜑) → ¬ 𝑤𝑅𝑦))
1312expdimp 453 . . . 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 2960 1 (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤𝐴 (𝑤𝑅𝑦[𝑤 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  [wsbc 3417   class class class wbr 4613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614
This theorem is referenced by:  bnj110  30636
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