Theorem iota4 5171

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 2017	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		biimpr 129	. . . . . 6 ⊢ ((𝜑 ↔ 𝑥 = 𝑧) → (𝑥 = 𝑧 → 𝜑))
3	2	alimi 1443	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑥(𝑥 = 𝑧 → 𝜑))
4		sb2 1755	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑧 → 𝜑) → [𝑧 / 𝑥]𝜑)
5	3, 4	syl 14	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → [𝑧 / 𝑥]𝜑)
6		iotaval 5164	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
7	6	eqcomd 2171	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑))
8		dfsbcq2 2954	. . . . 5 ⊢ (𝑧 = (℩𝑥𝜑) → ([𝑧 / 𝑥]𝜑 ↔ [(℩𝑥𝜑) / 𝑥]𝜑))
9	7, 8	syl 14	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ([𝑧 / 𝑥]𝜑 ↔ [(℩𝑥𝜑) / 𝑥]𝜑))
10	5, 9	mpbid 146	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑)
11	10	exlimiv 1586	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑)
12	1, 11	sylbi 120	1 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341   = wceq 1343  ∃wex 1480  [wsb 1750  ∃!weu 2014  [wsbc 2951  ℩cio 5151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-sn 3582  df-pr 3583  df-uni 3790  df-iota 5153

This theorem is referenced by:  iota4an  5172  iotacl  5176