Theorem eu6lem 2644

Description: Proof lines shared by eu6 2645 and eu6im 2646. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2640 was then proved as dfeu 2667. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.) Extract common proof lines. (Revised by Wolf Lammen, 3-Mar-2023.)

Assertion

Ref	Expression
eu6lem	⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)))

Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eu6lem

Step	Hyp	Ref	Expression
1		19.42v 2052	. . . 4 ⊢ (∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝑦 = 𝑧) ↔ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∃𝑧 𝑦 = 𝑧))
2		pm3.33 781	. . . . . . . . 9 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ (𝜑 → 𝑥 = 𝑧)) → (𝑥 = 𝑦 → 𝑥 = 𝑧))
3	2	alanimi 1915	. . . . . . . 8 ⊢ ((∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝜑 → 𝑥 = 𝑧)) → ∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧))
4		equvelv 2137	. . . . . . . 8 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) ↔ 𝑦 = 𝑧)
5	3, 4	sylib 210	. . . . . . 7 ⊢ ((∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝜑 → 𝑥 = 𝑧)) → 𝑦 = 𝑧)
6	5	pm4.71i 555	. . . . . 6 ⊢ ((∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝜑 → 𝑥 = 𝑧)) ↔ ((∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝜑 → 𝑥 = 𝑧)) ∧ 𝑦 = 𝑧))
7		albiim 1991	. . . . . . . . 9 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (∀𝑥(𝜑 → 𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
8	7	biancomi 456	. . . . . . . 8 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝜑 → 𝑥 = 𝑦)))
9		equequ2 2130	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
10	9	imbi2d 332	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝑧)))
11	10	albidv 2019	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑧)))
12	11	anbi2d 622	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝜑 → 𝑥 = 𝑦)) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝜑 → 𝑥 = 𝑧))))
13	8, 12	syl5bb 275	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝜑 → 𝑥 = 𝑧))))
14	13	pm5.32ri 571	. . . . . 6 ⊢ ((∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝑦 = 𝑧) ↔ ((∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝜑 → 𝑥 = 𝑧)) ∧ 𝑦 = 𝑧))
15	6, 14	bitr4i 270	. . . . 5 ⊢ ((∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝜑 → 𝑥 = 𝑧)) ↔ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝑦 = 𝑧))
16	15	exbii 1947	. . . 4 ⊢ (∃𝑧(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝜑 → 𝑥 = 𝑧)) ↔ ∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝑦 = 𝑧))
17		ax6evr 2119	. . . . 5 ⊢ ∃𝑧 𝑦 = 𝑧
18	17	biantru 525	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∃𝑧 𝑦 = 𝑧))
19	1, 16, 18	3bitr4ri 296	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑧(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝜑 → 𝑥 = 𝑧)))
20	19	exbii 1947	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑦∃𝑧(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝜑 → 𝑥 = 𝑧)))
21		exdistrv 2054	. 2 ⊢ (∃𝑦∃𝑧(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝜑 → 𝑥 = 𝑧)) ↔ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)))
22	20, 21	bitri 267	1 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1654  ∃wex 1878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879

This theorem is referenced by:  eu6  2645  eu6im  2646