Theorem spliteq-P6-L1 775

Description: Lemma for spliteq-P6-L1 775.

Assertion

Ref	Expression
spliteq-P6-L1	⊢ (𝑎 = 𝑡 → (𝑡 = 𝑢 ↔ ∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢)))

Distinct variable groups:   𝑡,𝑎   𝑢,𝑎

Proof of Theorem spliteq-P6-L1

Step	Hyp	Ref	Expression
1		rcp-NDASM1of2 193	. . . . 5 ⊢ ((𝑎 = 𝑡 ∧ 𝑡 = 𝑢) → 𝑎 = 𝑡)
2		eqtrns-P5.CL 631	. . . . 5 ⊢ ((𝑎 = 𝑡 ∧ 𝑡 = 𝑢) → 𝑎 = 𝑢)
3	1, 2	ndandi-P3.7 172	. . . 4 ⊢ ((𝑎 = 𝑡 ∧ 𝑡 = 𝑢) → (𝑎 = 𝑡 ∧ 𝑎 = 𝑢))
4		exi-P6 718	. . . 4 ⊢ ((𝑎 = 𝑡 ∧ 𝑎 = 𝑢) → ∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢))
5	3, 4	syl-P3.24.RC 260	. . 3 ⊢ ((𝑎 = 𝑡 ∧ 𝑡 = 𝑢) → ∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢))
6	5	rcp-NDIMI2 224	. 2 ⊢ (𝑎 = 𝑡 → (𝑡 = 𝑢 → ∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢)))
7		rcp-NDASM1of2 193	. . . . . 6 ⊢ ((𝑎 = 𝑡 ∧ 𝑎 = 𝑢) → 𝑎 = 𝑡)
8	7	eqsym-P5 627	. . . . 5 ⊢ ((𝑎 = 𝑡 ∧ 𝑎 = 𝑢) → 𝑡 = 𝑎)
9		rcp-NDASM2of2 194	. . . . 5 ⊢ ((𝑎 = 𝑡 ∧ 𝑎 = 𝑢) → 𝑎 = 𝑢)
10	8, 9	eqtrns-P5 630	. . . 4 ⊢ ((𝑎 = 𝑡 ∧ 𝑎 = 𝑢) → 𝑡 = 𝑢)
11	10	exiav-P5 615	. . 3 ⊢ (∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢) → 𝑡 = 𝑢)
12	11	rcp-NDIMP0addall 207	. 2 ⊢ (𝑎 = 𝑡 → (∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢) → 𝑡 = 𝑢))
13	6, 12	ndbii-P3.13 178	1 ⊢ (𝑎 = 𝑡 → (𝑡 = 𝑢 ↔ ∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢)))

Syntax hints:  term-obj 1   = wff-equals 6   → 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-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

This theorem is referenced by:  spliteq-P6  776