Theorem iota1 4353

Description: Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
iota1	⊢ (∃!xφ → (φ ↔ (℩xφ) = x))

Proof of Theorem iota1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2208	. 2 ⊢ (∃!xφ ↔ ∃z∀x(φ ↔ x = z))
2		sp 1747	. . . . 5 ⊢ (∀x(φ ↔ x = z) → (φ ↔ x = z))
3		iotaval 4350	. . . . . 6 ⊢ (∀x(φ ↔ x = z) → (℩xφ) = z)
4	3	eqeq2d 2364	. . . . 5 ⊢ (∀x(φ ↔ x = z) → (x = (℩xφ) ↔ x = z))
5	2, 4	bitr4d 247	. . . 4 ⊢ (∀x(φ ↔ x = z) → (φ ↔ x = (℩xφ)))
6		eqcom 2355	. . . 4 ⊢ (x = (℩xφ) ↔ (℩xφ) = x)
7	5, 6	syl6bb 252	. . 3 ⊢ (∀x(φ ↔ x = z) → (φ ↔ (℩xφ) = x))
8	7	exlimiv 1634	. 2 ⊢ (∃z∀x(φ ↔ x = z) → (φ ↔ (℩xφ) = x))
9	1, 8	sylbi 187	1 ⊢ (∃!xφ → (φ ↔ (℩xφ) = x))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642  ∃!weu 2204  ℩cio 4337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339

This theorem is referenced by:  iota2df  4365  sniota  4369