Theorem bnj23 35174

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 3793	. . . . 5 ⊢ (𝑤 ∈ V → ([𝑤 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑤 / 𝑥]𝜑))
2	1	elv 3462	. . . 4 ⊢ ([𝑤 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑤 / 𝑥]𝜑)
3		bnj23.1	. . . . . . . 8 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
4	3	eleq2i 2857	. . . . . . 7 ⊢ (𝑤 ∈ 𝐵 ↔ 𝑤 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})
5		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
6	5	elrabsf 3791	. . . . . . 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 5114	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑤𝑅𝑦))
9	8	notbid 321	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑤𝑅𝑦))
10	9	rspccv 3580	. . . . . 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 458	. . . 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 3159	1 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑦 → [𝑤 / 𝑥]𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  {crab 3418  Vcvv 3457  [wsbc 3746   class class class wbr 5111

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112

This theorem is used by:  bnj110  35313