Theorem iotavalb 45262

Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6511. (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 6511	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
2		iotasbc 45251	. . . 4 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ ∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦)))
3		iotaexeu 45250	. . . . 5 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
4		eqsbc1 3788	. . . . 5 ⊢ ((℩𝑥𝜑) ∈ V → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦))
5	3, 4	syl 18	. . . 4 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦))
6	2, 5	bitr3d 284	. . 3 ⊢ (∃!𝑥𝜑 → (∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
7		equequ2 2059	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
8	7	bibi2d 345	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑧) ↔ (𝜑 ↔ 𝑥 = 𝑦)))
9	8	albidv 1953	. . . . 5 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
10	9	biimpac 484	. . . 4 ⊢ ((∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
11	10	exlimiv 1963	. . 3 ⊢ (∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
12	6, 11	biimtrrdi 257	. 2 ⊢ (∃!𝑥𝜑 → ((℩𝑥𝜑) = 𝑦 → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
13	1, 12	impbid2 229	1 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃!weu 2595  Vcvv 3453  [wsbc 3742  ℩cio 6491

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-sbc 3743  df-un 3907  df-ss 3919  df-sn 4588  df-pr 4590  df-uni 4871  df-iota 6493

This theorem is used by:  iotavalsb  45265