Theorem euxfr2dc 3011

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
euxfr2dc.1	⊢ 𝐴 ∈ V
euxfr2dc.2	⊢ ∃*𝑦 𝑥 = 𝐴

Assertion

Ref	Expression
euxfr2dc	⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑))

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴

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

Proof of Theorem euxfr2dc

Step	Hyp	Ref	Expression
1		euxfr2dc.2	. . . . . . 7 ⊢ ∃*𝑦 𝑥 = 𝐴
2	1	moani 2157	. . . . . 6 ⊢ ∃*𝑦(𝜑 ∧ 𝑥 = 𝐴)
3		ancom 266	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝜑))
4	3	mobii 2123	. . . . . 6 ⊢ (∃*𝑦(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑))
5	2, 4	mpbi 145	. . . . 5 ⊢ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑)
6	5	ax-gen 1502	. . . 4 ⊢ ∀𝑥∃*𝑦(𝑥 = 𝐴 ∧ 𝜑)
7		excom 1716	. . . . . 6 ⊢ (∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑))
8	7	dcbii 852	. . . . 5 ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ DECID ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑))
9		2euswapdc 2178	. . . . 5 ⊢ (DECID ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) → (∀𝑥∃*𝑦(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑))))
10	8, 9	sylbi 121	. . . 4 ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∀𝑥∃*𝑦(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑))))
11	6, 10	mpi 15	. . 3 ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
12		moeq 3001	. . . . . . 7 ⊢ ∃*𝑥 𝑥 = 𝐴
13	12	moani 2157	. . . . . 6 ⊢ ∃*𝑥(𝜑 ∧ 𝑥 = 𝐴)
14	3	mobii 2123	. . . . . 6 ⊢ (∃*𝑥(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑))
15	13, 14	mpbi 145	. . . . 5 ⊢ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑)
16	15	ax-gen 1502	. . . 4 ⊢ ∀𝑦∃*𝑥(𝑥 = 𝐴 ∧ 𝜑)
17		2euswapdc 2178	. . . 4 ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∀𝑦∃*𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑))))
18	16, 17	mpi 15	. . 3 ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑)))
19	11, 18	impbid 129	. 2 ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
20		euxfr2dc.1	. . . 4 ⊢ 𝐴 ∈ V
21		biidd 172	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜑))
22	20, 21	ceqsexv 2861	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜑)
23	22	eubii 2095	. 2 ⊢ (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑)
24	19, 23	bitrdi 196	1 ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846  ∀wal 1400   = wceq 1402  ∃wex 1545  ∃!weu 2086  ∃*wmo 2087   ∈ wcel 2209  Vcvv 2821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823

This theorem is referenced by:  euxfrdc  3012