Theorem euxfr2w 3679

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr2 3681 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 14-Nov-2004.) Avoid ax-13 2377. (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 2631	. . . 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 2554	. . . . 5 ⊢ ∃*𝑦(𝜑 ∧ 𝑥 = 𝐴)
4		ancom 460	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝜑))
5	4	mobii 2549	. . . . 5 ⊢ (∃*𝑦(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑))
6	3, 5	mpbi 230	. . . 4 ⊢ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑)
7	1, 6	mpg 1799	. . 3 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
8		2euswapv 2631	. . . 4 ⊢ (∀𝑦∃*𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑)))
9		moeq 3666	. . . . . 6 ⊢ ∃*𝑥 𝑥 = 𝐴
10	9	moani 2554	. . . . 5 ⊢ ∃*𝑥(𝜑 ∧ 𝑥 = 𝐴)
11	4	mobii 2549	. . . . 5 ⊢ (∃*𝑥(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑))
12	10, 11	mpbi 230	. . . 4 ⊢ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑)
13	8, 12	mpg 1799	. . 3 ⊢ (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑))
14	7, 13	impbii 209	. 2 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
15		euxfr2w.1	. . . 4 ⊢ 𝐴 ∈ V
16		biidd 262	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜑))
17	15, 16	ceqsexv 3491	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜑)
18	17	eubii 2586	. 2 ⊢ (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑)
19	14, 18	bitri 275	1 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃*wmo 2538  ∃!weu 2569  Vcvv 3441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-mo 2540  df-eu 2570  df-cleq 2729  df-clel 2812

This theorem is referenced by:  euxfrw  3680  euop2  5461