Theorem 2eu8 2657

Description: Two equivalent expressions for double existential uniqueness. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥∃!𝑦 using 2eu7 2656. Usage of this theorem is discouraged because it depends on ax-13 2375. (Contributed by NM, 20-Feb-2005.) (New usage is discouraged.)

Assertion

Ref	Expression
2eu8	⊢ (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑))

Proof of Theorem 2eu8

Step	Hyp	Ref	Expression
1		2eu2 2651	. . 3 ⊢ (∃!𝑥∃𝑦𝜑 → (∃!𝑦∃!𝑥𝜑 ↔ ∃!𝑦∃𝑥𝜑))
2	1	pm5.32i 574	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))
3		nfeu1 2586	. . . . 5 ⊢ Ⅎ𝑥∃!𝑥𝜑
4	3	nfeu 2592	. . . 4 ⊢ Ⅎ𝑥∃!𝑦∃!𝑥𝜑
5	4	euan 2619	. . 3 ⊢ (∃!𝑥(∃!𝑦∃!𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃!𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑))
6		ancom 460	. . . . . 6 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃𝑦𝜑 ∧ ∃!𝑥𝜑))
7	6	eubii 2583	. . . . 5 ⊢ (∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑦(∃𝑦𝜑 ∧ ∃!𝑥𝜑))
8		nfe1 2148	. . . . . 6 ⊢ Ⅎ𝑦∃𝑦𝜑
9	8	euan 2619	. . . . 5 ⊢ (∃!𝑦(∃𝑦𝜑 ∧ ∃!𝑥𝜑) ↔ (∃𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑))
10		ancom 460	. . . . 5 ⊢ ((∃𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑦∃!𝑥𝜑 ∧ ∃𝑦𝜑))
11	7, 9, 10	3bitri 297	. . . 4 ⊢ (∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃!𝑥𝜑 ∧ ∃𝑦𝜑))
12	11	eubii 2583	. . 3 ⊢ (∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥(∃!𝑦∃!𝑥𝜑 ∧ ∃𝑦𝜑))
13		ancom 460	. . 3 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑦∃!𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑))
14	5, 12, 13	3bitr4ri 304	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑))
15		2eu7 2656	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))
16	2, 14, 15	3bitr3ri 302	1 ⊢ (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1776  ∃!weu 2566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-mo 2538  df-eu 2567

This theorem is referenced by: (None)