Theorem iotavalb 44970

Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6491. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotavalb	⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem iotavalb

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iotaval 6491	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
2		iotasbc 44959	. . . 4 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ ∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦)))
3		iotaexeu 44958	. . . . 5 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
4		eqsbc1 3790	. . . . 5 ⊢ ((℩𝑥𝜑) ∈ V → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦))
5	3, 4	syl 17	. . . 4 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦))
6	2, 5	bitr3d 283	. . 3 ⊢ (∃!𝑥𝜑 → (∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
7		equequ2 2045	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
8	7	bibi2d 344	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑧) ↔ (𝜑 ↔ 𝑥 = 𝑦)))
9	8	albidv 1939	. . . . 5 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
10	9	biimpac 482	. . . 4 ⊢ ((∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
11	10	exlimiv 1949	. . 3 ⊢ (∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
12	6, 11	biimtrrdi 256	. 2 ⊢ (∃!𝑥𝜑 → ((℩𝑥𝜑) = 𝑦 → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
13	1, 12	impbid2 228	1 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∃!weu 2594  Vcvv 3453  [wsbc 3744  ℩cio 6471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-sbc 3745  df-un 3909  df-ss 3921  df-sn 4582  df-pr 4584  df-uni 4865  df-iota 6473

This theorem is referenced by:  iotavalsb  44973