Theorem euxfr2 3022

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A. (Contributed by NM, 14-Nov-2004.)

Hypotheses

Ref	Expression
euxfr2.1	⊢ A ∈ V
euxfr2.2	⊢ ∃*y x = A

Assertion

Ref	Expression
euxfr2	⊢ (∃!x∃y(x = A ∧ φ) ↔ ∃!yφ)

Distinct variable groups:   φ,x   x,A

Allowed substitution hints:   φ(y)   A(y)

Proof of Theorem euxfr2

Step	Hyp	Ref	Expression
1		2euswap 2280	. . . 4 ⊢ (∀x∃*y(x = A ∧ φ) → (∃!x∃y(x = A ∧ φ) → ∃!y∃x(x = A ∧ φ)))
2		euxfr2.2	. . . . . 6 ⊢ ∃*y x = A
3	2	moani 2256	. . . . 5 ⊢ ∃*y(φ ∧ x = A)
4		ancom 437	. . . . . 6 ⊢ ((φ ∧ x = A) ↔ (x = A ∧ φ))
5	4	mobii 2240	. . . . 5 ⊢ (∃*y(φ ∧ x = A) ↔ ∃*y(x = A ∧ φ))
6	3, 5	mpbi 199	. . . 4 ⊢ ∃*y(x = A ∧ φ)
7	1, 6	mpg 1548	. . 3 ⊢ (∃!x∃y(x = A ∧ φ) → ∃!y∃x(x = A ∧ φ))
8		2euswap 2280	. . . 4 ⊢ (∀y∃*x(x = A ∧ φ) → (∃!y∃x(x = A ∧ φ) → ∃!x∃y(x = A ∧ φ)))
9		moeq 3013	. . . . . 6 ⊢ ∃*x x = A
10	9	moani 2256	. . . . 5 ⊢ ∃*x(φ ∧ x = A)
11	4	mobii 2240	. . . . 5 ⊢ (∃*x(φ ∧ x = A) ↔ ∃*x(x = A ∧ φ))
12	10, 11	mpbi 199	. . . 4 ⊢ ∃*x(x = A ∧ φ)
13	8, 12	mpg 1548	. . 3 ⊢ (∃!y∃x(x = A ∧ φ) → ∃!x∃y(x = A ∧ φ))
14	7, 13	impbii 180	. 2 ⊢ (∃!x∃y(x = A ∧ φ) ↔ ∃!y∃x(x = A ∧ φ))
15		euxfr2.1	. . . 4 ⊢ A ∈ V
16		biidd 228	. . . 4 ⊢ (x = A → (φ ↔ φ))
17	15, 16	ceqsexv 2895	. . 3 ⊢ (∃x(x = A ∧ φ) ↔ φ)
18	17	eubii 2213	. 2 ⊢ (∃!y∃x(x = A ∧ φ) ↔ ∃!yφ)
19	14, 18	bitri 240	1 ⊢ (∃!x∃y(x = A ∧ φ) ↔ ∃!yφ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  euxfr  3023