Theorem bnj1185 33835

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 5152	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤𝑅𝑧 ↔ 𝑦𝑅𝑧))
3	2	notbid 318	. . . . 5 ⊢ (𝑤 = 𝑦 → (¬ 𝑤𝑅𝑧 ↔ ¬ 𝑦𝑅𝑧))
4	3	cbvralvw 3235	. . . 4 ⊢ (∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)
5	4	rexbii 3095	. . 3 ⊢ (∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)
6	1, 5	sylib 217	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)
7		eleq1w 2817	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
8		breq2 5153	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝑥))
9	8	notbid 318	. . . . . 6 ⊢ (𝑧 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑦𝑅𝑥))
10	9	ralbidv 3178	. . . . 5 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
11	7, 10	anbi12d 632	. . . 4 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)))
12	11	cbvexvw 2041	. . 3 ⊢ (∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
13		df-rex 3072	. . 3 ⊢ (∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧))
14		df-rex 3072	. . 3 ⊢ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
15	12, 13, 14	3bitr4ri 304	. 2 ⊢ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)
16	6, 15	sylibr 233	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397  ∃wex 1782   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   class class class wbr 5149

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150

This theorem is referenced by:  bnj1190  34050  bnj1189  34051