Theorem bnj23 35052

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 = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}

Assertion

Ref	Expression
bnj23	⊢ (∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤 ∈ 𝐴 (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

Step	Hyp	Ref	Expression
1		sbcng 3800	. . . . 5 ⊢ (𝑤 ∈ V → ([𝑤 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑤 / 𝑥]𝜑))
2	1	elv 3468	. . . 4 ⊢ ([𝑤 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑤 / 𝑥]𝜑)
3		bnj23.1	. . . . . . . 8 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
4	3	eleq2i 2861	. . . . . . 7 ⊢ (𝑤 ∈ 𝐵 ↔ 𝑤 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})
5		nfcv 2931	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
6	5	elrabsf 3798	. . . . . . 7 ⊢ (𝑤 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥] ¬ 𝜑))
7	4, 6	bitri 278	. . . . . 6 ⊢ (𝑤 ∈ 𝐵 ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥] ¬ 𝜑))
8		breq1 5116	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑤𝑅𝑦))
9	8	notbid 321	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑤𝑅𝑦))
10	9	rspccv 3587	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 → (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑦))
11	7, 10	biimtrrid 246	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 → ((𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥] ¬ 𝜑) → ¬ 𝑤𝑅𝑦))
12	11	expdimp 457	. . . 4 ⊢ ((∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 ∧ 𝑤 ∈ 𝐴) → ([𝑤 / 𝑥] ¬ 𝜑 → ¬ 𝑤𝑅𝑦))
13	2, 12	biimtrrid 246	. . 3 ⊢ ((∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 ∧ 𝑤 ∈ 𝐴) → (¬ [𝑤 / 𝑥]𝜑 → ¬ 𝑤𝑅𝑦))
14	13	con4d 116	. 2 ⊢ ((∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 ∧ 𝑤 ∈ 𝐴) → (𝑤𝑅𝑦 → [𝑤 / 𝑥]𝜑))
15	14	ralrimiva 3163	1 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑦 → [𝑤 / 𝑥]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423  Vcvv 3463  [wsbc 3753   class class class wbr 5113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114

This theorem is referenced by:  bnj110  35191