Theorem iota4 6517

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		eu6 2574	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		biimpr 220	. . . . . 6 ⊢ ((𝜑 ↔ 𝑥 = 𝑧) → (𝑥 = 𝑧 → 𝜑))
3	2	alimi 1811	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑥(𝑥 = 𝑧 → 𝜑))
4		sb6 2086	. . . . 5 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑))
5	3, 4	sylibr 234	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → [𝑧 / 𝑥]𝜑)
6		iotaval 6507	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
7	6	eqcomd 2742	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑))
8		dfsbcq2 3773	. . . . 5 ⊢ (𝑧 = (℩𝑥𝜑) → ([𝑧 / 𝑥]𝜑 ↔ [(℩𝑥𝜑) / 𝑥]𝜑))
9	7, 8	syl 17	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ([𝑧 / 𝑥]𝜑 ↔ [(℩𝑥𝜑) / 𝑥]𝜑))
10	5, 9	mpbid 232	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑)
11	10	exlimiv 1930	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑)
12	1, 11	sylbi 217	1 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540  ∃wex 1779  [wsb 2065  ∃!weu 2568  [wsbc 3770  ℩cio 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-sbc 3771  df-un 3936  df-ss 3948  df-sn 4607  df-pr 4609  df-uni 4889  df-iota 6489

This theorem is referenced by:  iota4an  6518  iotacl  6522  pm14.24  44423  sbiota1  44425  eubrdm  47032