Theorem euxfr 3653

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 2372. Use the weaker euxfrw 3651 when possible. (Contributed by NM, 14-Nov-2004.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
euxfr	⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

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

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

Proof of Theorem euxfr

Step	Hyp	Ref	Expression
1		euxfr.2	. . . . . 6 ⊢ ∃!𝑦 𝑥 = 𝐴
2		euex 2577	. . . . . 6 ⊢ (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
3	1, 2	ax-mp 5	. . . . 5 ⊢ ∃𝑦 𝑥 = 𝐴
4	3	biantrur 530	. . . 4 ⊢ (𝜑 ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑))
5		19.41v 1954	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑))
6		euxfr.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	6	pm5.32i 574	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜓))
8	7	exbii 1851	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓))
9	4, 5, 8	3bitr2i 298	. . 3 ⊢ (𝜑 ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓))
10	9	eubii 2585	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜓))
11		euxfr.1	. . 3 ⊢ 𝐴 ∈ V
12	1	eumoi 2579	. . 3 ⊢ ∃*𝑦 𝑥 = 𝐴
13	11, 12	euxfr2 3652	. 2 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦𝜓)
14	10, 13	bitri 274	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∃!weu 2568  Vcvv 3422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-mo 2540  df-eu 2569  df-cleq 2730  df-clel 2817

This theorem is referenced by: (None)