Theorem euxfrdc 2778

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
euxfrdc.1	⊢ 𝐴 ∈ V
euxfrdc.2	⊢ ∃!𝑦 𝑥 = 𝐴
euxfrdc.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
euxfrdc	⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑦𝜓))

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

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

Proof of Theorem euxfrdc

Step	Hyp	Ref	Expression
1		euxfrdc.2	. . . . . 6 ⊢ ∃!𝑦 𝑥 = 𝐴
2		euex 1971	. . . . . 6 ⊢ (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
3	1, 2	ax-mp 7	. . . . 5 ⊢ ∃𝑦 𝑥 = 𝐴
4	3	biantrur 297	. . . 4 ⊢ (𝜑 ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑))
5		19.41v 1823	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑))
6		euxfrdc.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	6	pm5.32i 441	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜓))
8	7	exbii 1536	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓))
9	4, 5, 8	3bitr2i 206	. . 3 ⊢ (𝜑 ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓))
10	9	eubii 1950	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜓))
11		euxfrdc.1	. . 3 ⊢ 𝐴 ∈ V
12	1	eumoi 1974	. . 3 ⊢ ∃*𝑦 𝑥 = 𝐴
13	11, 12	euxfr2dc 2777	. 2 ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦𝜓))
14	10, 13	syl5bb 190	1 ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑦𝜓))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  DECID wdc 775   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∃!weu 1941  Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-dc 776  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by: (None)