Theorem euxfrdc 2912

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 2044	. . . . . 6 ⊢ (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
3	1, 2	ax-mp 5	. . . . 5 ⊢ ∃𝑦 𝑥 = 𝐴
4	3	biantrur 301	. . . 4 ⊢ (𝜑 ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑))
5		19.41v 1890	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑))
6		euxfrdc.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	6	pm5.32i 450	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜓))
8	7	exbii 1593	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓))
9	4, 5, 8	3bitr2i 207	. . 3 ⊢ (𝜑 ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓))
10	9	eubii 2023	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜓))
11		euxfrdc.1	. . 3 ⊢ 𝐴 ∈ V
12	1	eumoi 2047	. . 3 ⊢ ∃*𝑦 𝑥 = 𝐴
13	11, 12	euxfr2dc 2911	. 2 ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦𝜓))
14	10, 13	syl5bb 191	1 ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑦𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 824   = wceq 1343  ∃wex 1480  ∃!weu 2014   ∈ wcel 2136  Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by: (None)