Theorem pm14.24 44428

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 2586	. . . . 5 ⊢ Ⅎ𝑥∃!𝑥𝜑
2		nfsbc1v 3811	. . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		pm14.12 44417	. . . . . . . . . 10 ⊢ (∃!𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
4	3	19.21bbi 2188	. . . . . . . . 9 ⊢ (∃!𝑥𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
5	4	ancomsd 465	. . . . . . . 8 ⊢ (∃!𝑥𝜑 → (([𝑦 / 𝑥]𝜑 ∧ 𝜑) → 𝑥 = 𝑦))
6	5	expdimp 452	. . . . . . 7 ⊢ ((∃!𝑥𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜑 → 𝑥 = 𝑦))
7		pm13.13b 44404	. . . . . . . . 9 ⊢ (([𝑦 / 𝑥]𝜑 ∧ 𝑥 = 𝑦) → 𝜑)
8	7	ex 412	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦 → 𝜑))
9	8	adantl 481	. . . . . . 7 ⊢ ((∃!𝑥𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝑥 = 𝑦 → 𝜑))
10	6, 9	impbid 212	. . . . . 6 ⊢ ((∃!𝑥𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜑 ↔ 𝑥 = 𝑦))
11	10	ex 412	. . . . 5 ⊢ (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → (𝜑 ↔ 𝑥 = 𝑦)))
12	1, 2, 11	alrimd 2213	. . . 4 ⊢ (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
13		iotaval 6534	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
14	13	eqcomd 2741	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
15	12, 14	syl6 35	. . 3 ⊢ (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → 𝑦 = (℩𝑥𝜑)))
16		iota4 6544	. . . 4 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
17		dfsbcq 3793	. . . 4 ⊢ (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜑 ↔ [(℩𝑥𝜑) / 𝑥]𝜑))
18	16, 17	syl5ibrcom 247	. . 3 ⊢ (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) → [𝑦 / 𝑥]𝜑))
19	15, 18	impbid 212	. 2 ⊢ (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = (℩𝑥𝜑)))
20	19	alrimiv 1925	1 ⊢ (∃!𝑥𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = (℩𝑥𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃!weu 2566  [wsbc 3791  ℩cio 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480  df-sbc 3792  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516

This theorem is referenced by: (None)