Theorem eqmiddle-P6 708

Description: Introduce Intermediate Middle Variable from Equality.

'𝑦' cannot occur in either '𝛾' or '𝑡'.

Hypothesis

Ref	Expression
eqmiddle-P6.1	⊢ (𝛾 → 𝑥 = 𝑡)

Assertion

Ref	Expression
eqmiddle-P6	⊢ (𝛾 → ∃𝑦(𝑥 = 𝑦 ∧ 𝑦 = 𝑡))

Distinct variable groups:   𝛾,𝑦   𝑡,𝑦   𝑥,𝑦

Proof of Theorem eqmiddle-P6

Step	Hyp	Ref	Expression
1		axL6ex-P5 625	. . 3 ⊢ ∃𝑦 𝑦 = 𝑡
2	1	rcp-NDIMP0addall 207	. 2 ⊢ (𝛾 → ∃𝑦 𝑦 = 𝑡)
3		eqmiddle-P6.1	. . . . . . 7 ⊢ (𝛾 → 𝑥 = 𝑡)
4	3	rcp-NDIMP1add1 208	. . . . . 6 ⊢ ((𝛾 ∧ 𝑦 = 𝑡) → 𝑥 = 𝑡)
5		rcp-NDASM2of2 194	. . . . . . 7 ⊢ ((𝛾 ∧ 𝑦 = 𝑡) → 𝑦 = 𝑡)
6	5	subeqr-P5 635	. . . . . 6 ⊢ ((𝛾 ∧ 𝑦 = 𝑡) → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡))
7	4, 6	bimpr-P4 533	. . . . 5 ⊢ ((𝛾 ∧ 𝑦 = 𝑡) → 𝑥 = 𝑦)
8	7, 5	ndandi-P3.7 172	. . . 4 ⊢ ((𝛾 ∧ 𝑦 = 𝑡) → (𝑥 = 𝑦 ∧ 𝑦 = 𝑡))
9	8	rcp-NDIMI2 224	. . 3 ⊢ (𝛾 → (𝑦 = 𝑡 → (𝑥 = 𝑦 ∧ 𝑦 = 𝑡)))
10	9	alloverimex-P5.GENV 622	. 2 ⊢ (𝛾 → (∃𝑦 𝑦 = 𝑡 → ∃𝑦(𝑥 = 𝑦 ∧ 𝑦 = 𝑡)))
11	2, 10	ndime-P3.6 171	1 ⊢ (𝛾 → ∃𝑦(𝑥 = 𝑦 ∧ 𝑦 = 𝑡))

Syntax hints:  term-obj 1   = wff-equals 6   → wff-imp 10   ∧ 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

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:  eqmiddle-P6.CL  709