Theorem euxfr2 3358

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 2536	. . . 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 2513	. . . . 5 ⊢ ∃*𝑦(𝜑 ∧ 𝑥 = 𝐴)
4		ancom 465	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝜑))
5	4	mobii 2481	. . . . 5 ⊢ (∃*𝑦(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑))
6	3, 5	mpbi 219	. . . 4 ⊢ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑)
7	1, 6	mpg 1715	. . 3 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
8		2euswap 2536	. . . 4 ⊢ (∀𝑦∃*𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑)))
9		moeq 3349	. . . . . 6 ⊢ ∃*𝑥 𝑥 = 𝐴
10	9	moani 2513	. . . . 5 ⊢ ∃*𝑥(𝜑 ∧ 𝑥 = 𝐴)
11	4	mobii 2481	. . . . 5 ⊢ (∃*𝑥(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑))
12	10, 11	mpbi 219	. . . 4 ⊢ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑)
13	8, 12	mpg 1715	. . 3 ⊢ (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑))
14	7, 13	impbii 198	. 2 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
15		euxfr2.1	. . . 4 ⊢ 𝐴 ∈ V
16		biidd 251	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜑))
17	15, 16	ceqsexv 3215	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜑)
18	17	eubii 2480	. 2 ⊢ (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑)
19	14, 18	bitri 263	1 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  ∃*wmo 2459  Vcvv 3173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175

This theorem is referenced by:  euxfr  3359  euop2  4899