Theorem euxfr2 3692

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Usage of this theorem is discouraged because it depends on ax-13 2410. Use the weaker euxfr2w 3690 when possible. (Contributed by NM, 14-Nov-2004.) (New usage is discouraged.)

Hypotheses

Ref	Expression
euxfr2.1	⊢ 𝐴 ∈ V
euxfr2.2	⊢ ∃*𝑦 𝑥 = 𝐴

Assertion

Ref	Expression
euxfr2	⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑)

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴

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

Proof of Theorem euxfr2

Step	Hyp	Ref	Expression
1		2euswap 2679	. . . 4 ⊢ (∀𝑥∃*𝑦(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
2		euxfr2.2	. . . . . 6 ⊢ ∃*𝑦 𝑥 = 𝐴
3	2	moani 2587	. . . . 5 ⊢ ∃*𝑦(𝜑 ∧ 𝑥 = 𝐴)
4		ancom 465	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝜑))
5	4	mobii 2582	. . . . 5 ⊢ (∃*𝑦(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑))
6	3, 5	mpbi 233	. . . 4 ⊢ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑)
7	1, 6	mpg 1824	. . 3 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
8		2euswap 2679	. . . 4 ⊢ (∀𝑦∃*𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑)))
9		moeq 3677	. . . . . 6 ⊢ ∃*𝑥 𝑥 = 𝐴
10	9	moani 2587	. . . . 5 ⊢ ∃*𝑥(𝜑 ∧ 𝑥 = 𝐴)
11	4	mobii 2582	. . . . 5 ⊢ (∃*𝑥(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑))
12	10, 11	mpbi 233	. . . 4 ⊢ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑)
13	8, 12	mpg 1824	. . 3 ⊢ (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑))
14	7, 13	impbii 212	. 2 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
15		euxfr2.1	. . . 4 ⊢ 𝐴 ∈ V
16		biidd 265	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜑))
17	15, 16	ceqsexv 3509	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜑)
18	17	eubii 2619	. 2 ⊢ (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑)
19	14, 18	bitri 278	1 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃*wmo 2571  ∃!weu 2602  Vcvv 3461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-13 2410  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-mo 2573  df-eu 2603  df-cleq 2761  df-clel 2844

This theorem is referenced by:  euxfr  3693