Theorem spliteq-P6 776

Description: Split Equality Into Left and Right Halves.

'𝑎' is distinct from all other variables.

Assertion

Ref	Expression
spliteq-P6	⊢ (𝑡 = 𝑢 ↔ ∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢))

Distinct variable groups:   𝑡,𝑎   𝑢,𝑎

Proof of Theorem spliteq-P6

Step	Hyp	Ref	Expression
1		axL6ex-P5 625	. 2 ⊢ ∃𝑎 𝑎 = 𝑡
2		nfrv-P6 686	. . . 4 ⊢ Ⅎ𝑎 𝑡 = 𝑢
3		nfrex1-P6 742	. . . 4 ⊢ Ⅎ𝑎∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢)
4	2, 3	nfrbi-P6 691	. . 3 ⊢ Ⅎ𝑎(𝑡 = 𝑢 ↔ ∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢))
5		spliteq-P6-L1 775	. . 3 ⊢ (𝑎 = 𝑡 → (𝑡 = 𝑢 ↔ ∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢)))
6	4, 5	exia-P6 746	. 2 ⊢ (∃𝑎 𝑎 = 𝑡 → (𝑡 = 𝑢 ↔ ∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢)))
7	1, 6	rcp-NDIME0 228	1 ⊢ (𝑡 = 𝑢 ↔ ∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢))

Syntax hints:  term-obj 1   = wff-equals 6   ↔ 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-L10 27  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:  psubspliteq-P6  800