Theorem splitelof-P6 778

Description: Split "Element Of" Predicate Into Left and Right Halves.

'𝑎' and '𝑏' are distinct from all other variables.

Assertion

Ref	Expression
splitelof-P6	⊢ (𝑡 ∈ 𝑢 ↔ ∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏))

Distinct variable groups:   𝑡,𝑎   𝑢,𝑎   𝑡,𝑏   𝑢,𝑏,𝑎

Proof of Theorem splitelof-P6

Step	Hyp	Ref	Expression
1		axL6ex-P5 625	. 2 ⊢ ∃𝑏 𝑏 = 𝑢
2		nfrv-P6 686	. . . 4 ⊢ Ⅎ𝑏 𝑡 ∈ 𝑢
3		nfrex1-P6 742	. . . . 5 ⊢ Ⅎ𝑏∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏)
4	3	nfrex2-P6 744	. . . 4 ⊢ Ⅎ𝑏∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏)
5	2, 4	nfrbi-P6 691	. . 3 ⊢ Ⅎ𝑏(𝑡 ∈ 𝑢 ↔ ∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏))
6		axL6ex-P5 625	. . . 4 ⊢ ∃𝑎 𝑎 = 𝑡
7		nfrv-P6 686	. . . . . 6 ⊢ Ⅎ𝑎 𝑏 = 𝑢
8		nfrv-P6 686	. . . . . . 7 ⊢ Ⅎ𝑎 𝑡 ∈ 𝑢
9		nfrex1-P6 742	. . . . . . 7 ⊢ Ⅎ𝑎∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏)
10	8, 9	nfrbi-P6 691	. . . . . 6 ⊢ Ⅎ𝑎(𝑡 ∈ 𝑢 ↔ ∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏))
11	7, 10	nfrim-P6 689	. . . . 5 ⊢ Ⅎ𝑎(𝑏 = 𝑢 → (𝑡 ∈ 𝑢 ↔ ∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏)))
12		splitelof-P6-L1 777	. . . . . 6 ⊢ ((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) → (𝑡 ∈ 𝑢 ↔ ∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏)))
13	12	rcp-NDIMI2 224	. . . . 5 ⊢ (𝑎 = 𝑡 → (𝑏 = 𝑢 → (𝑡 ∈ 𝑢 ↔ ∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏))))
14	11, 13	exia-P6 746	. . . 4 ⊢ (∃𝑎 𝑎 = 𝑡 → (𝑏 = 𝑢 → (𝑡 ∈ 𝑢 ↔ ∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏))))
15	6, 14	rcp-NDIME0 228	. . 3 ⊢ (𝑏 = 𝑢 → (𝑡 ∈ 𝑢 ↔ ∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏)))
16	5, 15	exia-P6 746	. 2 ⊢ (∃𝑏 𝑏 = 𝑢 → (𝑡 ∈ 𝑢 ↔ ∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏)))
17	1, 16	rcp-NDIME0 228	1 ⊢ (𝑡 ∈ 𝑢 ↔ ∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏))

Syntax hints:  term-obj 1   = wff-equals 6   ∈ wff-elemof 7   → wff-imp 10   ↔ wff-bi 104   ∧ wff-and 132  ∃wff-exists 595

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19  ax-L8-inl 20  ax-L8-inr 21  ax-L10 27  ax-L11 28  ax-L12 29

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682

This theorem is referenced by:  psubsplitelof-P6  801