Theorem exists1 1434

Description: Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru 2740.

Assertion

Ref	Expression
exists1	⊢ (∃!x x = x ↔ ∀x x = y)

Distinct variable group:   x,y

Proof of Theorem exists1

Step	Hyp	Ref	Expression
1		df-eu 1359	. 2 ⊢ (∃!x x = x ↔ ∃y∀x(x = x ↔ x = y))
2		equid 1113	. . . . . 6 ⊢ x = x
3	2	tbt 717	. . . . 5 ⊢ (x = y ↔ (x = y ↔ x = x))
4		bicom 518	. . . . 5 ⊢ ((x = y ↔ x = x) ↔ (x = x ↔ x = y))
5	3, 4	bitr 173	. . . 4 ⊢ (x = y ↔ (x = x ↔ x = y))
6	5	albii 975	. . 3 ⊢ (∀x x = y ↔ ∀x(x = x ↔ x = y))
7	6	exbii 1027	. 2 ⊢ (∃y∀x x = y ↔ ∃y∀x(x = x ↔ x = y))
8		hbae 1128	. . 3 ⊢ (∀x x = y → ∀y∀x x = y)
9	8	19.9 1012	. 2 ⊢ (∃y∀x x = y ↔ ∀x x = y)
10	1, 7, 9	3bitr2 179	1 ⊢ (∃!x x = x ↔ ∀x x = y)

Syntax hints:   ↔ wb 146  ∀wal 950  ∃wex 956   = wceq 1099  ∃!weu 1357

This theorem is referenced by:  exists2 1435

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-eu 1359

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