Theorem euxfr2 1923

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

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

Proof of Theorem euxfr2

Step	Hyp	Ref	Expression
1		2euswap 1444	. . . 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 1422	. . . . 5 ⊢ ∃*y(φ ⋀ x = A)
4		ancom 435	. . . . . 6 ⊢ ((φ ⋀ x = A) ↔ (x = A ⋀ φ))
5	4	mobii 1404	. . . . 5 ⊢ (∃*y(φ ⋀ x = A) ↔ ∃*y(x = A ⋀ φ))
6	3, 5	mpbi 189	. . . 4 ⊢ ∃*y(x = A ⋀ φ)
7	1, 6	mpg 985	. . 3 ⊢ (∃!x∃y(x = A ⋀ φ) → ∃!y∃x(x = A ⋀ φ))
8		2euswap 1444	. . . 4 ⊢ (∀y∃*x(x = A ⋀ φ) → (∃!y∃x(x = A ⋀ φ) → ∃!x∃y(x = A ⋀ φ)))
9		moeq 1917	. . . . . 6 ⊢ ∃*x x = A
10	9	moani 1422	. . . . 5 ⊢ ∃*x(φ ⋀ x = A)
11	4	mobii 1404	. . . . 5 ⊢ (∃*x(φ ⋀ x = A) ↔ ∃*x(x = A ⋀ φ))
12	10, 11	mpbi 189	. . . 4 ⊢ ∃*x(x = A ⋀ φ)
13	8, 12	mpg 985	. . 3 ⊢ (∃!y∃x(x = A ⋀ φ) → ∃!x∃y(x = A ⋀ φ))
14	7, 13	impbi 157	. 2 ⊢ (∃!x∃y(x = A ⋀ φ) ↔ ∃!y∃x(x = A ⋀ φ))
15		euxfr2.1	. . . 4 ⊢ A ∈ V
16		pm4.2i 171	. . . 4 ⊢ (x = A → (φ ↔ φ))
17	15, 16	ceqsexv 1832	. . 3 ⊢ (∃x(x = A ⋀ φ) ↔ φ)
18	17	eubii 1386	. 2 ⊢ (∃!y∃x(x = A ⋀ φ) ↔ ∃!yφ)
19	14, 18	bitr 173	1 ⊢ (∃!x∃y(x = A ⋀ φ) ↔ ∃!yφ)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 955   ∈ wcel 957  ∃wex 979  ∃!weu 1379  ∃*wmo 1380  Vcvv 1808

This theorem is referenced by:  euxfr 1924  euop2 2802

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809

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