Theorem iota4 4357

Description: Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
iota4	⊢ (∃!xφ → [̣(℩xφ) / x]̣φ)

Proof of Theorem iota4

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		bi2 189	. . . . . 6 ⊢ ((φ ↔ x = z) → (x = z → φ))
3	2	alimi 1559	. . . . 5 ⊢ (∀x(φ ↔ x = z) → ∀x(x = z → φ))
4		sb2 2023	. . . . 5 ⊢ (∀x(x = z → φ) → [z / x]φ)
5	3, 4	syl 15	. . . 4 ⊢ (∀x(φ ↔ x = z) → [z / x]φ)
6		iotaval 4350	. . . . . 6 ⊢ (∀x(φ ↔ x = z) → (℩xφ) = z)
7	6	eqcomd 2358	. . . . 5 ⊢ (∀x(φ ↔ x = z) → z = (℩xφ))
8		dfsbcq2 3049	. . . . 5 ⊢ (z = (℩xφ) → ([z / x]φ ↔ [̣(℩xφ) / x]̣φ))
9	7, 8	syl 15	. . . 4 ⊢ (∀x(φ ↔ x = z) → ([z / x]φ ↔ [̣(℩xφ) / x]̣φ))
10	5, 9	mpbid 201	. . 3 ⊢ (∀x(φ ↔ x = z) → [̣(℩xφ) / x]̣φ)
11	10	exlimiv 1634	. 2 ⊢ (∃z∀x(φ ↔ x = z) → [̣(℩xφ) / x]̣φ)
12	1, 11	sylbi 187	1 ⊢ (∃!xφ → [̣(℩xφ) / x]̣φ)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648  ∃!weu 2204  [̣wsbc 3046  ℩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:  iota4an  4358  iotacl  4362