Metamath Proof Explorer


Theorem bnj1185

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

Ref Expression
Hypothesis bnj1185.1 ( 𝜑 → ∃ 𝑧𝐵𝑤𝐵 ¬ 𝑤 𝑅 𝑧 )
Assertion bnj1185 ( 𝜑 → ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )

Proof

Step Hyp Ref Expression
1 bnj1185.1 ( 𝜑 → ∃ 𝑧𝐵𝑤𝐵 ¬ 𝑤 𝑅 𝑧 )
2 breq1 ( 𝑤 = 𝑦 → ( 𝑤 𝑅 𝑧𝑦 𝑅 𝑧 ) )
3 2 notbid ( 𝑤 = 𝑦 → ( ¬ 𝑤 𝑅 𝑧 ↔ ¬ 𝑦 𝑅 𝑧 ) )
4 3 cbvralvw ( ∀ 𝑤𝐵 ¬ 𝑤 𝑅 𝑧 ↔ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑧 )
5 4 rexbii ( ∃ 𝑧𝐵𝑤𝐵 ¬ 𝑤 𝑅 𝑧 ↔ ∃ 𝑧𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑧 )
6 1 5 sylib ( 𝜑 → ∃ 𝑧𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑧 )
7 eleq1w ( 𝑧 = 𝑥 → ( 𝑧𝐵𝑥𝐵 ) )
8 breq2 ( 𝑧 = 𝑥 → ( 𝑦 𝑅 𝑧𝑦 𝑅 𝑥 ) )
9 8 notbid ( 𝑧 = 𝑥 → ( ¬ 𝑦 𝑅 𝑧 ↔ ¬ 𝑦 𝑅 𝑥 ) )
10 9 ralbidv ( 𝑧 = 𝑥 → ( ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑧 ↔ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑥 ) )
11 7 10 anbi12d ( 𝑧 = 𝑥 → ( ( 𝑧𝐵 ∧ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑧 ) ↔ ( 𝑥𝐵 ∧ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑥 ) ) )
12 11 cbvexvw ( ∃ 𝑧 ( 𝑧𝐵 ∧ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑧 ) ↔ ∃ 𝑥 ( 𝑥𝐵 ∧ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑥 ) )
13 df-rex ( ∃ 𝑧𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑧 ↔ ∃ 𝑧 ( 𝑧𝐵 ∧ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑧 ) )
14 df-rex ( ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 ↔ ∃ 𝑥 ( 𝑥𝐵 ∧ ∀ 𝑦𝐵 ¬ 𝑦 𝑅 𝑥 ) )
15 12 13 14 3bitr4ri ( ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 ↔ ∃ 𝑧𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑧 )
16 6 15 sylibr ( 𝜑 → ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )