Theorem euxfr2w 3685

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr2 3687 with a disjoint variable condition, which does not require ax-13 2405. (Contributed by NM, 14-Nov-2004.) Avoid ax-13 2405. (Revised by GG, 10-Jan-2024.)

Hypotheses

Ref	Expression
euxfr2w.1	⊢ 𝐴 ∈ V
euxfr2w.2	⊢ ∃*𝑦 𝑥 = 𝐴

Assertion

Ref	Expression
euxfr2w	⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑)

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

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

Proof of Theorem euxfr2w

Step	Hyp	Ref	Expression
1		2euswapv 2659	. . . 4 ⊢ (∀𝑥∃*𝑦(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
2		euxfr2w.2	. . . . . 6 ⊢ ∃*𝑦 𝑥 = 𝐴
3	2	moani 2582	. . . . 5 ⊢ ∃*𝑦(𝜑 ∧ 𝑥 = 𝐴)
4		ancom 464	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝜑))
5	4	mobii 2577	. . . . 5 ⊢ (∃*𝑦(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑))
6	3, 5	mpbi 232	. . . 4 ⊢ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑)
7	1, 6	mpg 1819	. . 3 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
8		2euswapv 2659	. . . 4 ⊢ (∀𝑦∃*𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑)))
9		moeq 3672	. . . . . 6 ⊢ ∃*𝑥 𝑥 = 𝐴
10	9	moani 2582	. . . . 5 ⊢ ∃*𝑥(𝜑 ∧ 𝑥 = 𝐴)
11	4	mobii 2577	. . . . 5 ⊢ (∃*𝑥(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑))
12	10, 11	mpbi 232	. . . 4 ⊢ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑)
13	8, 12	mpg 1819	. . 3 ⊢ (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑))
14	7, 13	impbii 211	. 2 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
15		euxfr2w.1	. . . 4 ⊢ 𝐴 ∈ V
16		biidd 264	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜑))
17	15, 16	ceqsexv 3504	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜑)
18	17	eubii 2614	. 2 ⊢ (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑)
19	14, 18	bitri 277	1 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562  ∃wex 1801   ∈ wcel 2144  ∃*wmo 2566  ∃!weu 2597  Vcvv 3456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-nf 1806  df-mo 2568  df-eu 2598  df-cleq 2756  df-clel 2839

This theorem is referenced by:  euxfrw  3686  euop2  5483