Theorem bnj1185 35289

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1185.1	⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧)

Assertion

Ref	Expression
bnj1185	⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)

Distinct variable groups:   𝑤,𝐵,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑤,𝑅,𝑦,𝑧   𝑥,𝑅

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧, 𝑤)

Proof of Theorem bnj1185

Step	Hyp	Ref	Expression
1		bnj1185.1	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧)
2		breq1 5110	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤𝑅𝑧 ↔ 𝑦𝑅𝑧))
3	2	notbid 321	. . . . 5 ⊢ (𝑤 = 𝑦 → (¬ 𝑤𝑅𝑧 ↔ ¬ 𝑦𝑅𝑧))
4	3	cbvralvw 3242	. . . 4 ⊢ (∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)
5	4	rexbii 3111	. . 3 ⊢ (∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)
6	1, 5	sylib 221	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)
7		eleq1w 2845	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
8		breq2 5111	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝑥))
9	8	notbid 321	. . . . . 6 ⊢ (𝑧 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑦𝑅𝑥))
10	9	ralbidv 3187	. . . . 5 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
11	7, 10	anbi12d 644	. . . 4 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)))
12	11	cbvexvw 2070	. . 3 ⊢ (∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
13		df-rex 3089	. . 3 ⊢ (∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧))
14		df-rex 3089	. . 3 ⊢ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
15	12, 13, 14	3bitr4ri 307	. 2 ⊢ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)
16	6, 15	sylibr 237	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∃wex 1812   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088   class class class wbr 5107

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 2147  ax-9 2155  ax-ext 2734

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-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108

This theorem is used by:  bnj1190  35504  bnj1189  35505