Theorem euxfr 3023

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
euxfr.1	⊢ A ∈ V
euxfr.2	⊢ ∃!y x = A
euxfr.3	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
euxfr	⊢ (∃!xφ ↔ ∃!yψ)

Distinct variable groups:   ψ,x   φ,y   x,A

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

Proof of Theorem euxfr

Step	Hyp	Ref	Expression
1		euxfr.2	. . . . . 6 ⊢ ∃!y x = A
2		euex 2227	. . . . . 6 ⊢ (∃!y x = A → ∃y x = A)
3	1, 2	ax-mp 5	. . . . 5 ⊢ ∃y x = A
4	3	biantrur 492	. . . 4 ⊢ (φ ↔ (∃y x = A ∧ φ))
5		19.41v 1901	. . . 4 ⊢ (∃y(x = A ∧ φ) ↔ (∃y x = A ∧ φ))
6		euxfr.3	. . . . . 6 ⊢ (x = A → (φ ↔ ψ))
7	6	pm5.32i 618	. . . . 5 ⊢ ((x = A ∧ φ) ↔ (x = A ∧ ψ))
8	7	exbii 1582	. . . 4 ⊢ (∃y(x = A ∧ φ) ↔ ∃y(x = A ∧ ψ))
9	4, 5, 8	3bitr2i 264	. . 3 ⊢ (φ ↔ ∃y(x = A ∧ ψ))
10	9	eubii 2213	. 2 ⊢ (∃!xφ ↔ ∃!x∃y(x = A ∧ ψ))
11		euxfr.1	. . 3 ⊢ A ∈ V
12	1	eumoi 2245	. . 3 ⊢ ∃*y x = A
13	11, 12	euxfr2 3022	. 2 ⊢ (∃!x∃y(x = A ∧ ψ) ↔ ∃!yψ)
14	10, 13	bitri 240	1 ⊢ (∃!xφ ↔ ∃!yψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  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: (None)