Theorem euxfrdc 2727

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 1930	. . . . . 6 ⊢ (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
3	1, 2	ax-mp 7	. . . . 5 ⊢ ∃𝑦 𝑥 = 𝐴
4	3	biantrur 287	. . . 4 ⊢ (𝜑 ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑))
5		19.41v 1782	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑))
6		euxfrdc.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	6	pm5.32i 427	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜓))
8	7	exbii 1496	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓))
9	4, 5, 8	3bitr2i 197	. . 3 ⊢ (𝜑 ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓))
10	9	eubii 1909	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜓))
11		euxfrdc.1	. . 3 ⊢ 𝐴 ∈ V
12	1	eumoi 1933	. . 3 ⊢ ∃*𝑦 𝑥 = 𝐴
13	11, 12	euxfr2dc 2726	. 2 ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦𝜓))
14	10, 13	syl5bb 181	1 ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑦𝜓))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  DECID wdc 742   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃!weu 1900  Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-dc 743  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by: (None)