Theorem euxfr 1923

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A.

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

Proof of Theorem euxfr

Step	Hyp	Ref	Expression
1		euxfr.2	. . . . . 6 ⊢ ∃!y x = A
2		euex 1392	. . . . . 6 ⊢ (∃!y x = A → ∃y x = A)
3	1, 2	ax-mp 7	. . . . 5 ⊢ ∃y x = A
4	3	biantrur 724	. . . 4 ⊢ (φ ↔ (∃y x = A ⋀ φ))
5		19.41v 1303	. . . 4 ⊢ (∃y(x = A ⋀ φ) ↔ (∃y x = A ⋀ φ))
6		euxfr.3	. . . . . 6 ⊢ (x = A → (φ ↔ ψ))
7	6	pm5.32i 644	. . . . 5 ⊢ ((x = A ⋀ φ) ↔ (x = A ⋀ ψ))
8	7	exbii 1049	. . . 4 ⊢ (∃y(x = A ⋀ φ) ↔ ∃y(x = A ⋀ ψ))
9	4, 5, 8	3bitr2 179	. . 3 ⊢ (φ ↔ ∃y(x = A ⋀ ψ))
10	9	eubii 1385	. 2 ⊢ (∃!xφ ↔ ∃!x∃y(x = A ⋀ ψ))
11		euxfr.1	. . 3 ⊢ A ∈ V
12	1	eumoi 1410	. . 3 ⊢ ∃*y x = A
13	11, 12	euxfr2 1922	. 2 ⊢ (∃!x∃y(x = A ⋀ ψ) ↔ ∃!yψ)
14	10, 13	bitr 173	1 ⊢ (∃!xφ ↔ ∃!yψ)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 954   ∈ wcel 956  ∃wex 978  ∃!weu 1378  Vcvv 1807

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808

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