Theorem pm14.24 45510

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 2614	. . . . 5 ⊢ Ⅎ𝑥∃!𝑥𝜑
2		nfsbc1v 3758	. . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		pm14.12 45499	. . . . . . . . . 10 ⊢ (∃!𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
4	3	19.21bbi 2226	. . . . . . . . 9 ⊢ (∃!𝑥𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
5	4	ancomsd 471	. . . . . . . 8 ⊢ (∃!𝑥𝜑 → (([𝑦 / 𝑥]𝜑 ∧ 𝜑) → 𝑥 = 𝑦))
6	5	expdimp 458	. . . . . . 7 ⊢ ((∃!𝑥𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜑 → 𝑥 = 𝑦))
7		pm13.13b 45486	. . . . . . . . 9 ⊢ (([𝑦 / 𝑥]𝜑 ∧ 𝑥 = 𝑦) → 𝜑)
8	7	ex 418	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦 → 𝜑))
9	8	adantl 487	. . . . . . 7 ⊢ ((∃!𝑥𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝑥 = 𝑦 → 𝜑))
10	6, 9	impbid 215	. . . . . 6 ⊢ ((∃!𝑥𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜑 ↔ 𝑥 = 𝑦))
11	10	ex 418	. . . . 5 ⊢ (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → (𝜑 ↔ 𝑥 = 𝑦)))
12	1, 2, 11	alrimd 2251	. . . 4 ⊢ (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
13		iotaval 6512	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
14	13	eqcomd 2766	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
15	12, 14	syl6 36	. . 3 ⊢ (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → 𝑦 = (℩𝑥𝜑)))
16		iota4 6519	. . . 4 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
17		dfsbcq 3740	. . . 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 1959	1 ⊢ (∃!𝑥𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = (℩𝑥𝜑)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569  ∃!weu 2593  [wsbc 3738  ℩cio 6492

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-v 3452  df-sbc 3739  df-ss 3896  df-un 4151  df-sn 4585  df-pr 4587  df-uni 4867  df-iota 6494

This theorem is used by: (None)