Theorem iotavalb 42048

Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6407. (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 6407	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
2		iotasbc 42037	. . . 4 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ ∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦)))
3		iotaexeu 42036	. . . . 5 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
4		eqsbc1 3765	. . . . 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 280	. . 3 ⊢ (∃!𝑥𝜑 → (∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
7		equequ2 2029	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
8	7	bibi2d 343	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑧) ↔ (𝜑 ↔ 𝑥 = 𝑦)))
9	8	albidv 1923	. . . . 5 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
10	9	biimpac 479	. . . 4 ⊢ ((∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
11	10	exlimiv 1933	. . 3 ⊢ (∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
12	6, 11	syl6bir 253	. 2 ⊢ (∃!𝑥𝜑 → ((℩𝑥𝜑) = 𝑦 → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
13	1, 12	impbid2 225	1 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃!weu 2568  Vcvv 3432  [wsbc 3716  ℩cio 6389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-sbc 3717  df-un 3892  df-in 3894  df-ss 3904  df-sn 4562  df-pr 4564  df-uni 4840  df-iota 6391

This theorem is referenced by:  iotavalsb  42051