Metamath Proof Explorer


Theorem bnj1189

Description: Technical lemma for bnj69 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1189.1 ( 𝜑 ↔ ( 𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅ ) )
bnj1189.2 ( 𝜓 ↔ ( 𝑥𝐵𝑦𝐵𝑦 𝑅 𝑥 ) )
bnj1189.3 ( 𝜒 ↔ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )
Assertion bnj1189 ( 𝜑 → ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )

Proof

Step Hyp Ref Expression
1 bnj1189.1 ( 𝜑 ↔ ( 𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅ ) )
2 bnj1189.2 ( 𝜓 ↔ ( 𝑥𝐵𝑦𝐵𝑦 𝑅 𝑥 ) )
3 bnj1189.3 ( 𝜒 ↔ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )
4 n0 ( 𝐵 ≠ ∅ ↔ ∃ 𝑥 𝑥𝐵 )
5 4 biimpi ( 𝐵 ≠ ∅ → ∃ 𝑥 𝑥𝐵 )
6 1 5 bnj837 ( 𝜑 → ∃ 𝑥 𝑥𝐵 )
7 6 ancli ( 𝜑 → ( 𝜑 ∧ ∃ 𝑥 𝑥𝐵 ) )
8 19.42v ( ∃ 𝑥 ( 𝜑𝑥𝐵 ) ↔ ( 𝜑 ∧ ∃ 𝑥 𝑥𝐵 ) )
9 7 8 sylibr ( 𝜑 → ∃ 𝑥 ( 𝜑𝑥𝐵 ) )
10 3simpc ( ( 𝜑𝑥𝐵𝜒 ) → ( 𝑥𝐵𝜒 ) )
11 3 anbi2i ( ( 𝑥𝐵𝜒 ) ↔ ( 𝑥𝐵 ∧ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑥 ) )
12 10 11 sylib ( ( 𝜑𝑥𝐵𝜒 ) → ( 𝑥𝐵 ∧ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑥 ) )
13 12 19.8ad ( ( 𝜑𝑥𝐵𝜒 ) → ∃ 𝑥 ( 𝑥𝐵 ∧ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑥 ) )
14 df-rex ( ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 ↔ ∃ 𝑥 ( 𝑥𝐵 ∧ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑥 ) )
15 13 14 sylibr ( ( 𝜑𝑥𝐵𝜒 ) → ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )
16 15 3comr ( ( 𝜒𝜑𝑥𝐵 ) → ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )
17 16 3expib ( 𝜒 → ( ( 𝜑𝑥𝐵 ) → ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 ) )
18 simp1 ( ( 𝜑𝑥𝐵 ∧ ¬ 𝜒 ) → 𝜑 )
19 simp2 ( ( 𝜑𝑥𝐵 ∧ ¬ 𝜒 ) → 𝑥𝐵 )
20 rexnal ( ∃ 𝑦𝐵 ¬ ¬ 𝑦 𝑅 𝑥 ↔ ¬ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )
21 20 bicomi ( ¬ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑥 ↔ ∃ 𝑦𝐵 ¬ ¬ 𝑦 𝑅 𝑥 )
22 21 3 xchnxbir ( ¬ 𝜒 ↔ ∃ 𝑦𝐵 ¬ ¬ 𝑦 𝑅 𝑥 )
23 notnotb ( 𝑦 𝑅 𝑥 ↔ ¬ ¬ 𝑦 𝑅 𝑥 )
24 23 rexbii ( ∃ 𝑦𝐵 𝑦 𝑅 𝑥 ↔ ∃ 𝑦𝐵 ¬ ¬ 𝑦 𝑅 𝑥 )
25 22 24 bitr4i ( ¬ 𝜒 ↔ ∃ 𝑦𝐵 𝑦 𝑅 𝑥 )
26 25 biimpi ( ¬ 𝜒 → ∃ 𝑦𝐵 𝑦 𝑅 𝑥 )
27 26 bnj1196 ( ¬ 𝜒 → ∃ 𝑦 ( 𝑦𝐵𝑦 𝑅 𝑥 ) )
28 27 3ad2ant3 ( ( 𝜑𝑥𝐵 ∧ ¬ 𝜒 ) → ∃ 𝑦 ( 𝑦𝐵𝑦 𝑅 𝑥 ) )
29 3anass ( ( 𝑥𝐵𝑦𝐵𝑦 𝑅 𝑥 ) ↔ ( 𝑥𝐵 ∧ ( 𝑦𝐵𝑦 𝑅 𝑥 ) ) )
30 29 exbii ( ∃ 𝑦 ( 𝑥𝐵𝑦𝐵𝑦 𝑅 𝑥 ) ↔ ∃ 𝑦 ( 𝑥𝐵 ∧ ( 𝑦𝐵𝑦 𝑅 𝑥 ) ) )
31 19.42v ( ∃ 𝑦 ( 𝑥𝐵 ∧ ( 𝑦𝐵𝑦 𝑅 𝑥 ) ) ↔ ( 𝑥𝐵 ∧ ∃ 𝑦 ( 𝑦𝐵𝑦 𝑅 𝑥 ) ) )
32 30 31 bitri ( ∃ 𝑦 ( 𝑥𝐵𝑦𝐵𝑦 𝑅 𝑥 ) ↔ ( 𝑥𝐵 ∧ ∃ 𝑦 ( 𝑦𝐵𝑦 𝑅 𝑥 ) ) )
33 19 28 32 sylanbrc ( ( 𝜑𝑥𝐵 ∧ ¬ 𝜒 ) → ∃ 𝑦 ( 𝑥𝐵𝑦𝐵𝑦 𝑅 𝑥 ) )
34 33 2 bnj1198 ( ( 𝜑𝑥𝐵 ∧ ¬ 𝜒 ) → ∃ 𝑦 𝜓 )
35 19.42v ( ∃ 𝑦 ( 𝜑𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑦 𝜓 ) )
36 18 34 35 sylanbrc ( ( 𝜑𝑥𝐵 ∧ ¬ 𝜒 ) → ∃ 𝑦 ( 𝜑𝜓 ) )
37 1 2 bnj1190 ( ( 𝜑𝜓 ) → ∃ 𝑤𝐵𝑧𝐵 ¬ 𝑧 𝑅 𝑤 )
38 36 37 bnj593 ( ( 𝜑𝑥𝐵 ∧ ¬ 𝜒 ) → ∃ 𝑦𝑤𝐵𝑧𝐵 ¬ 𝑧 𝑅 𝑤 )
39 38 bnj937 ( ( 𝜑𝑥𝐵 ∧ ¬ 𝜒 ) → ∃ 𝑤𝐵𝑧𝐵 ¬ 𝑧 𝑅 𝑤 )
40 39 bnj1185 ( ( 𝜑𝑥𝐵 ∧ ¬ 𝜒 ) → ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )
41 40 3comr ( ( ¬ 𝜒𝜑𝑥𝐵 ) → ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )
42 41 3expib ( ¬ 𝜒 → ( ( 𝜑𝑥𝐵 ) → ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 ) )
43 17 42 pm2.61i ( ( 𝜑𝑥𝐵 ) → ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )
44 9 43 bnj593 ( 𝜑 → ∃ 𝑥𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )
45 nfre1 𝑥𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥
46 45 19.9 ( ∃ 𝑥𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 ↔ ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )
47 44 46 sylib ( 𝜑 → ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )