Theorem euxfr2w 3677

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr2 3679 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by NM, 14-Nov-2004.) Avoid ax-13 2401. (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 2655	. . . 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 2578	. . . . 5 ⊢ ∃*𝑦(𝜑 ∧ 𝑥 = 𝐴)
4		ancom 466	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝜑))
5	4	mobii 2573	. . . . 5 ⊢ (∃*𝑦(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑))
6	3, 5	mpbi 233	. . . 4 ⊢ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑)
7	1, 6	mpg 1830	. . 3 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
8		2euswapv 2655	. . . 4 ⊢ (∀𝑦∃*𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑)))
9		moeq 3664	. . . . . 6 ⊢ ∃*𝑥 𝑥 = 𝐴
10	9	moani 2578	. . . . 5 ⊢ ∃*𝑥(𝜑 ∧ 𝑥 = 𝐴)
11	4	mobii 2573	. . . . 5 ⊢ (∃*𝑥(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑))
12	10, 11	mpbi 233	. . . 4 ⊢ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑)
13	8, 12	mpg 1830	. . 3 ⊢ (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑))
14	7, 13	impbii 212	. 2 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
15		euxfr2w.1	. . . 4 ⊢ 𝐴 ∈ V
16		biidd 265	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜑))
17	15, 16	ceqsexv 3498	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜑)
18	17	eubii 2610	. 2 ⊢ (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑)
19	14, 18	bitri 278	1 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃*wmo 2562  ∃!weu 2593  Vcvv 3450

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-mo 2564  df-eu 2594  df-cleq 2752  df-clel 2835

This theorem is used by:  euxfrw  3678  euop2  5481