Theorem axregprim 35940

Description: ax-reg 9504 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)

Assertion

Ref	Expression
axregprim	⊢ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))

Proof of Theorem axregprim

Step	Hyp	Ref	Expression
1		axregnd 10525	. 2 ⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
2		df-an 397	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)) ↔ ¬ (𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
3	2	exbii 1855	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)) ↔ ∃𝑥 ¬ (𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
4		exnal 1834	. . 3 ⊢ (∃𝑥 ¬ (𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)) ↔ ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
5	3, 4	bitri 276	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)) ↔ ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
6	1, 5	sylib 219	1 ⊢ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-13 2380  ax-ext 2712  ax-sep 5225  ax-reg 9504

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-cleq 2732  df-clel 2815  df-nfc 2889

This theorem is referenced by: (None)