Theorem pm14.24 41588

Description: Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
pm14.24	⊢ (∃!𝑥𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = (℩𝑥𝜑)))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem pm14.24

Step	Hyp	Ref	Expression
1		nfeu1 2589	. . . . 5 ⊢ Ⅎ𝑥∃!𝑥𝜑
2		nfsbc1v 3700	. . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		pm14.12 41577	. . . . . . . . . 10 ⊢ (∃!𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
4	3	19.21bbi 2191	. . . . . . . . 9 ⊢ (∃!𝑥𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
5	4	ancomsd 469	. . . . . . . 8 ⊢ (∃!𝑥𝜑 → (([𝑦 / 𝑥]𝜑 ∧ 𝜑) → 𝑥 = 𝑦))
6	5	expdimp 456	. . . . . . 7 ⊢ ((∃!𝑥𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜑 → 𝑥 = 𝑦))
7		pm13.13b 41564	. . . . . . . . 9 ⊢ (([𝑦 / 𝑥]𝜑 ∧ 𝑥 = 𝑦) → 𝜑)
8	7	ex 416	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦 → 𝜑))
9	8	adantl 485	. . . . . . 7 ⊢ ((∃!𝑥𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝑥 = 𝑦 → 𝜑))
10	6, 9	impbid 215	. . . . . 6 ⊢ ((∃!𝑥𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜑 ↔ 𝑥 = 𝑦))
11	10	ex 416	. . . . 5 ⊢ (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → (𝜑 ↔ 𝑥 = 𝑦)))
12	1, 2, 11	alrimd 2217	. . . 4 ⊢ (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
13		iotaval 6313	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
14	13	eqcomd 2744	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
15	12, 14	syl6 35	. . 3 ⊢ (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → 𝑦 = (℩𝑥𝜑)))
16		iota4 6320	. . . 4 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
17		dfsbcq 3682	. . . 4 ⊢ (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜑 ↔ [(℩𝑥𝜑) / 𝑥]𝜑))
18	16, 17	syl5ibrcom 250	. . 3 ⊢ (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) → [𝑦 / 𝑥]𝜑))
19	15, 18	impbid 215	. 2 ⊢ (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = (℩𝑥𝜑)))
20	19	alrimiv 1934	1 ⊢ (∃!𝑥𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = (℩𝑥𝜑)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1540   = wceq 1542  ∃!weu 2569  [wsbc 3680  ℩cio 6295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-v 3400  df-sbc 3681  df-un 3848  df-in 3850  df-ss 3860  df-sn 4517  df-pr 4519  df-uni 4797  df-iota 6297

This theorem is referenced by: (None)