Theorem bnj23 32206

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 3744	. . . . 5 ⊢ (𝑤 ∈ V → ([𝑤 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑤 / 𝑥]𝜑))
2	1	elv 3416	. . . 4 ⊢ ([𝑤 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑤 / 𝑥]𝜑)
3		bnj23.1	. . . . . . . 8 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
4	3	eleq2i 2844	. . . . . . 7 ⊢ (𝑤 ∈ 𝐵 ↔ 𝑤 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})
5		nfcv 2920	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
6	5	elrabsf 3742	. . . . . . 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 5033	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑤𝑅𝑦))
9	8	notbid 322	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑤𝑅𝑦))
10	9	rspccv 3539	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 → (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑦))
11	7, 10	syl5bir 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	syl5bir 246	. . 3 ⊢ ((∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 ∧ 𝑤 ∈ 𝐴) → (¬ [𝑤 / 𝑥]𝜑 → ¬ 𝑤𝑅𝑦))
14	13	con4d 115	. 2 ⊢ ((∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 ∧ 𝑤 ∈ 𝐴) → (𝑤𝑅𝑦 → [𝑤 / 𝑥]𝜑))
15	14	ralrimiva 3114	1 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑦 → [𝑤 / 𝑥]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071  {crab 3075  Vcvv 3410  [wsbc 3697   class class class wbr 5030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rab 3080  df-v 3412  df-sbc 3698  df-un 3864  df-sn 4521  df-pr 4523  df-op 4527  df-br 5031

This theorem is referenced by:  bnj110  32348